A002531 -id:A002531

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A146983

a(n) = A002531(n)*A002531(n+1).

+20
3

1, 2, 10, 35, 133, 494, 1846, 6887, 25705, 95930, 358018, 1336139, 4986541, 18610022, 69453550, 259204175, 967363153, 3610248434, 13473630586, 50284273907, 187663465045, 700369586270, 2613814880038, 9754889933879, 36405744855481, 135868089488042

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

a(n+1) is the Hankel transform of A051960 aerated.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,3,-1).

FORMULA

From Peter Bala, May 01 2012: (Start)

a(n) = (-1)^n + 3*Sum_{k = 1..n} (-1)^(n-k)*6^(k-1)*binomial(n+k,2*k).

a(n) = (-1)^n*R(n,-3), where R(n,x) is the n-th row polynomial of A211955.

a(n) = (-1)^n*1/u*T(n,u)*T(n+1,u) with u = sqrt(-1/2) and T(n,x) denotes the Chebyshev polynomial of the first kind Cf. A182432.

Recurrence: a(n) = 4*a(n-1) -a(n-2) +3*(-1)^n, with a(0) = 1 and a(1) = 2; a(n)*a(n-2) = a(n-1)*(a(n-1)+3*(-1)^n).

Sum_{k>=0} (-1)^k/a(k) = 1/sqrt(3). (End)

From Colin Barker, Jul 29 2013: (Start)

a(n) = 3*a(n-1) + 3*a(n-2) - a(n-3).

G.f.: (1-x+x^2)/((1+x)*(1-4*x+x^2)). (End)

MAPLE

seq(coeff(series((1-x+x^2)/((1+x)*(1-4*x+x^2)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 09 2020

MATHEMATICA

LinearRecurrence[{3, 3, -1}, {1, 2, 10}, 30] (* G. C. Greubel, Jan 09 2020 *)

PROG

(PARI) my(x='x+O('x^30)); Vec((1-x+x^2)/((1+x)*(1-4*x+x^2))) \\ G. C. Greubel, Jan 09 2020

(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x+x^2)/((1+x)*(1-4*x+x^2)) )); // G. C. Greubel, Jan 09 2020

(Sage)

def A146983_list(prec):

P.<x> = PowerSeriesRing(ZZ, prec)

return P( (1-x+x^2)/((1+x)*(1-4*x+x^2)) ).list()

A146983_list(30) # G. C. Greubel, Jan 09 2020

(GAP) a:=[1, 2, 10];; for n in [4..30] do a[n]:=3*a[n-1]+3*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Jan 09 2020

CROSSREFS

Cf. A001654, A182432, A211955.

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Nov 04 2008

EXTENSIONS

More terms from Colin Barker, Jul 29 2013

STATUS

approved

A001353

a(n) = 4*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.
(Formerly M3499 N1420)

+10
188

0, 1, 4, 15, 56, 209, 780, 2911, 10864, 40545, 151316, 564719, 2107560, 7865521, 29354524, 109552575, 408855776, 1525870529, 5694626340, 21252634831, 79315912984, 296011017105, 1104728155436, 4122901604639, 15386878263120, 57424611447841, 214311567528244

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

3*a(n)^2 + 1 is a square. Moreover, 3*a(n)^2 + 1 = (2*a(n) - a(n-1))^2.

Consecutive terms give nonnegative solutions to x^2 - 4*x*y + y^2 = 1. - Max Alekseyev, Dec 12 2012

Values y solving the Pellian x^2 - 3*y^2 = 1; corresponding x values given by A001075(n). Moreover, we have a(n) = 2*a(n-1) + A001075(n-1). - Lekraj Beedassy, Jul 13 2006

Number of spanning trees in 2 X n grid: by examining what happens at the right-hand end we see that a(n) = 3*a(n-1) + 2*a(n-2) + 2*a(n-3) + ... + 2*a(1) + 1, where the final 1 corresponds to the tree ==...=| !. Solving this we get a(n) = 4*a(n-1) - a(n-2).

Complexity of 2 X n grid.

A016064 also describes triangles whose sides are consecutive integers and in which an inscribed circle has an integer radius. A001353 is exactly and precisely mapped to the integer radii of such inscribed circles, i.e., for each term of A016064, the corresponding term of A001353 gives the radius of the inscribed circle. - Harvey P. Dale, Dec 28 2000

n such that 3*n^2 = floor(sqrt(3)*n*ceiling(sqrt(3)*n)). - Benoit Cloitre, May 10 2003

For n>0, ratios a(n+1)/a(n) may be obtained as convergents of the continued fraction expansion of 2+sqrt(3): either as successive convergents of [4;-4] or as odd convergents of [3;1, 2]. - Lekraj Beedassy, Sep 19 2003

Ways of packing a 3 X (2*n-1) rectangle with dominoes, after attaching an extra square to the end of one of the sides of length 3. With reference to A001835, therefore: a(n) = a(n-1) + A001835(n-1) and A001835(n) = 3*A001835(n-1) + 2*a(n-1). - Joshua Zucker and the Castilleja School Math Club, Oct 28 2003

a(n+1) is a Chebyshev transform of 4^n, where the sequence with g.f. G(x) is sent to the sequence with g.f. (1/(1+x^2))G(x/(1+x^2)). - Paul Barry, Oct 25 2004

This sequence is prime-free, because a(2n) = a(n) * (a(n+1)-a(n-1)) and a(2n+1) = a(n+1)^2 - a(n)^2 = (a(n+1)+a(n)) * (a(n+1)-a(n)). - Jianing Song, Jul 06 2019

Numbers such that there is an m with t(n+m) = 3*t(m), where t(n) are the triangular numbers A000217. For instance, t(35) = 3*t(20) = 630, so 35 - 20 = 15 is in the sequence. - Floor van Lamoen, Oct 13 2005

a(n) = number of distinct matrix products in (A + B + C + D)^n where commutator [A,B] = 0 but neither A nor B commutes with C or D. - Paul D. Hanna and Max Alekseyev, Feb 01 2006

For n > 1, middle side (or long leg) of primitive Pythagorean triangles having an angle nearing Pi/3 with larger values of sides. [Complete triple (X, Y, Z), X < Y < Z, is given by X = A120892(n), Y = a(n), Z = A120893(n), with recurrence relations X(i+1) = 2*{X(i) - (-1)^i} + a(i); Z(i+1) = 2*{Z(i) + a(i)} - (-1)^i.] - Lekraj Beedassy, Jul 13 2006

From Dennis P. Walsh, Oct 04 2006: (Start)

Number of 2 X n simple rectangular mazes. A simple rectangular m X n maze is a graph G with vertex set {0, 1, ..., m} X {0, 1, ..., n} that satisfies the following two properties: (i) G consists of two orthogonal trees; (ii) one tree has a path that sequentially connects (0,0),(0,1), ..., (0,n), (1,n), ...,(m-1,n) and the other tree has a path that sequentially connects (1,0), (2,0), ..., (m,0), (m,1), ..., (m,n). For example, a(2) = 4 because there are four 2 X 2 simple rectangular mazes:

__ __ __ __

| | | |__ | | | | __|

| __| | __| | |__| | __|

(End)

[1, 4, 15, 56, 209, ...] is the Hankel transform of [1, 1, 5, 26, 139, 758, ...](see A005573). - Philippe Deléham, Apr 14 2007

The upper principal convergents to 3^(1/2), beginning with 2/1, 7/4, 26/15, 97/56, comprise a strictly decreasing sequence; numerators=A001075, denominators=A001353. - Clark Kimberling, Aug 27 2008

From Gary W. Adamson, Jun 21 2009: (Start)

A001353 and A001835 = bisection of continued fraction [1, 2, 1, 2, 1, 2, ...], i.e., of [1, 3, 4, 11, 15, 41, ...].

For n>0, a(n) equals the determinant of an (n-1) X (n-1) tridiagonal matrix with ones in the super and subdiagonals and (4, 4, 4, ...) as the main diagonal. [Corrected by Johannes Boot, Sep 04 2011]

A001835 and A001353 = right and next to right borders of triangle A125077. (End)

a(n) is equal to the permanent of the (n-1) X (n-1) Hessenberg matrix with 4's along the main diagonal, i's along the superdiagonal and the subdiagonal (i is the imaginary unit), and 0's everywhere else. - John M. Campbell, Jun 09 2011

2a(n) is the number of n-color compositions of 2n consisting of only even parts; see Guo in references. - Brian Hopkins, Jul 19 2011

Pisano period lengths: 1, 2, 6, 4, 3, 6, 8, 4, 18, 6, 10, 12, 12, 8, 6, 8, 18, 18, 5, 12, ... - R. J. Mathar, Aug 10 2012

From Michel Lagneau, Jul 08 2014: (Start)

a(n) is defined also by the recurrence a(1)=1; for n>1, a(n+1) = 2*a(n) + sqrt(3*a(n)^2 + 1) where a(n) is an integer for every n. This sequence is generalizable by the sequence b(n,m) of parameter m with the initial condition b(1,m) = 1, and for n > 1 b(n+1,m) = m*b(n,m) + sqrt((m^2 - 1)*b(n,m)^2 + 1) for m = 2, 3, 4, ... where b(n,m) is an integer for every n.

The first corresponding sequences are

b(n,2) = a(n) = A001353(n);

b(n,3) = A001109(n);

b(n,4) = A001090(n);

b(n,5) = A004189(n);

b(n,6) = A004191(n);

b(n,7) = A007655(n);

b(n,8) = A077412(n);

b(n,9) = A049660(n);

b(n,10) = A075843(n);

b(n,11) = A077421(n);

....................

We obtain a general sequence of polynomials {b(n,x)} = {1, 2*x, 4*x^2 - 1, 8*x^3 - 4*x, 16*x^4 - 12*x^2 + 1, 32*x^5 - 32*x^3 + 6*x, ...} with x = m where each b(n,x) is a Gegenbauer polynomial defined by the recurrence b(n,x)- 2*x*b(n-1,x) + b(n-2,x) = 0, the same relation as the Chebyshev recurrence, but with the initial conditions b(x,0) = 1 and b(x,1) = 2*x instead b(x,0) = 1 and b(x,1) = x for the Chebyshev polynomials. (End)

If a(n) denotes the n-th term of the above sequence and we construct a triangle whose sides are a(n) - 1, a(n) + 1 and sqrt(3a(n)^2 + 1), then, for every n the measure of one of the angles of the triangle so constructed will always be 120 degrees. This result of ours was published in Mathematics Spectrum (2012/2013), Vol. 45, No. 3, pp. 126-128. - K. S. Bhanu and Dr. M. N. Deshpande, Professor (Retd), Department of Statistics, Institute of Science, Nagpur (India).

For n >= 1, a(n) equals the number of 01-avoiding words of length n - 1 on alphabet {0, 1, 2, 3}. - Milan Janjic, Jan 25 2015

For n > 0, 10*a(n) is the number of vertices and roots on level n of the {4, 5} mosaic (see L. Németh Table 1 p. 6). - Michel Marcus, Oct 30 2015

(2 + sqrt(3))^n = A001075(n) + a(n)*sqrt(3), n >= 0; integers in the quadratic number field Q(sqrt(3)). - Wolfdieter Lang, Feb 16 2018

A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - Michael Somos, Dec 12 2019

The Cholesky decomposition A = C C* for tridiagonal A with A[i,i] = 4 and A[i+1,i] = A[i,i+1] = -1, as it arises in the discretized 2D Laplace operator (Poisson equation...), has nonzero elements C[i,i] = sqrt(a(i+1)/a(i)) = -1/C[i+1,i], i = 1, 2, 3, ... - M. F. Hasler, Mar 12 2021

The triples (a(n-1), 2a(n), a(n+1)), n=2,3,..., are exactly the triples (a,b,c) of positive integers a < b < c in arithmetic progression such that a*b+1, b*c+1, and c*a+1 are perfect squares. - Bernd Mulansky, Jul 10 2021

REFERENCES

J. Austin and L. Schneider, Generalized Fibonacci sequences in Pythagorean triple preserving sequences, Fib. Q., 58:1 (2020), 340-350.

Bastida, Julio R., Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163-166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009)

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; p. 163.

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 329.

J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 104.

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..200

Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.

Christian Aebi and Grant Cairns, Lattice Equable Parallelograms, arXiv:2006.07566 [math.NT], 2020.

Christian Aebi and Grant Cairns, Equable Parallelograms on the Eisenstein Lattice, arXiv:2401.08827 [math.CO], 2024. See p. 14.

W. K. Alt, Enumeration of Domino Tilings on the Projective Grid Graph, A Thesis Presented to The Division of Mathematics and Natural Sciences, Reed College, May 2013.

K. Andersen, L. Carbone, and D. Penta, Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.

Francesca Arici and Jens Kaad, Gysin sequences and SU(2)-symmetries of C*-algebras, arXiv:2012.11186 [math.OA], 2020.

Krassimir T. Atanassov and Anthony G. Shannon, On intercalated Fibonacci sequences, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 218-223.

Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, Linear recurrence sequences with indices in arithmetic progression and their sums, arXiv preprint arXiv:1505.06339 [math.NT], 2015.

D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 12.

K. S. Bhanu and M. N. Deshpande, Integral triangles with 120° angle Mathematics Spectrum, 45 (3) (2012/2013), 126-128.

Latham Boyle and Paul J. Steinhardt, Self-Similar One-Dimensional Quasilattices, arXiv preprint arXiv:1608.08220 [math-ph], 2016.

Fabrizio Canfora, Maxim Kurkov, Luigi Rosa, and Patrizia Vitale, The Gribov problem in Noncommutative QED, arXiv preprint arXiv:1505.06342 [hep-th], 2015.

Niccolò Castronuovo, On the number of fixed points of the map gamma, arXiv:2102.02739 [math.NT], 2021. Mentions this sequence.

Z. Cinkir, Effective Resistances, Kirchhoff index and Admissible Invariants of Ladder Graphs, arXiv preprint arXiv:1503.06353 [math.CO], 2015.

J. B. Cosgrave and K. Dilcher, A role for generalized Fermat numbers, Math. Comp. 86 (2017), 899-933 (see also paper #10).

M. N. Deshpande, One Interesting Family of Diophantine Triplets, International Journal of Mathematical Education In Science and Technology, Vol. 33 (No. 2, Mar-Apr), 2002.

M. N. Deshpande, Hansruedi Widmer and Zachary Franco, Simultaneous Squares from Arithmetic Progressions: 10622, The American Mathematical Monthly Vol. 106, No. 9 (Nov., 1999), 867-868.

Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607.

G. Dresden and Y. Li, Periodic Weighted Sums of Binomial Coefficients, arXiv:2210.04322 [math.NT], 2022.

E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.

F. Faase, Counting Hamiltonian cycles in product graphs

F. Faase, Results from the counting program

Felix Flicker, Time quasilattices in dissipative dynamical systems, arXiv:1707.09371 [nlin.CD], 2017. Also SciPost Phys. 5, 001 (2018).

D. Fortin, B-spline Toeplitz Inverse Under Corner Perturbations, International Journal of Pure and Applied Mathematics, Volume 77, No. 1, 2012, 107-118. - From N. J. A. Sloane, Oct 22 2012

Dale Gerdemann, Fractal images from (4, -1) recursion, YouTube, Oct 27 2014.

Juan B. Gil and Jessica A. Tomasko, Fibonacci colored compositions and applications, arXiv:2108.06462 [math.CO], 2021.

Andrew Granville and Zhi-Wei Sun, Values of Bernoulli polynomials, Pacific J. Math. 172 (1996), 117-137, at p. 119.

T. N. E. Greville, Table for third-degree spline interpolations with equally spaced arguments, Math. Comp., 24 (1970), 179-183.

Y.-H. Guo, n-Colour even self-inverse compositions, Proc. Indian Acad. Sci. (Math. Sci.), 120 (2010), 27-33.

B. Hopkins, Spotted tilings and n-color compositions, INTEGERS 12B (2012/2013), #A6.

A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=4, q=-1.

W. D. Hoskins, Table for third-degree spline interpolation using equi-spaced knots, Math. Comp., 25 (1971), 797-801.

Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

Tanya Khovanova, Recursive Sequences

Seong Ju Kim, R. Stees, and L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4

Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.

Germain Kreweras, Complexite et circuits Euleriens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212.

Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, Fausto Jarquín-Zárate, The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d)), arXiv:1904.13002 [math.NT], 2019.

Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq.(44), lhs, m=6.

Ioana-Claudia Lazăr, Lucas sequences in t-uniform simplicial complexes, arXiv:1904.06555 [math.GR], 2019.

Hojoo Lee, Problems in elementary number theory Problem I 18.

E. Keith Lloyd, The Standard Deviation of 1, 2,..., n: Pell's Equation and Rational Triangles, Math. Gaz. vol 81 (1997), 231-243.

Dino Lorenzini and Z. Xiang, Integral points on variable separated curves, Preprint 2016.

Valcho Milchev and Tsvetelina Karamfilova, Domino tiling in grid - new dependence, arXiv:1707.09741 [math.HO], 2017.

László Németh, Trees on hyperbolic honeycombs, arXiv:1510.08311 [math.CO], 2015.

Hideyuki Ohtskua, proposer, Problem B-1351, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 62, No. 3 (2024), p. 258.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.

Ariel D. Procaccia and Jamie Tucker-Foltz, Compact Redistricting Plans Have Many Spanning Trees, Harvard Univ. (2021).

P. Raff, Analysis of the Number of Spanning Trees of K_2 x P_n. Contains sequence, recurrence, generating function, and more. [From Paul Raff, Mar 06 2009]

Ryan Stees, Sequences of Spiral Knot Determinants, Senior Honors Projects, Paper 84, James Madison Univ., May 2016.

D. P. Walsh, Counting n x 2 Simple Rectangular Mazes

F. V. Waugh and M. W. Maxfield, Side-and-diagonal numbers, Math. Mag., 40 (1967), 74-83.

Eric Weisstein's World of Mathematics, Ladder Graph

Eric Weisstein's World of Mathematics, Spanning Tree

Jianqiang Zhao, Finite Multiple zeta Values and Finite Euler Sums, arXiv preprint arXiv:1507.04917 [math.NT], 2015.

Index entries for linear recurrences with constant coefficients, signature (4,-1)

Index entries for sequences related to Chebyshev polynomials.

FORMULA

G.f.: x/(1-4*x+x^2).

a(n) = ((2 + sqrt(3))^n - (2 - sqrt(3))^n)/(2*sqrt(3)).

a(n) = sqrt((A001075(n)^2 - 1)/3).

a(n) = 2*a(n-1) + sqrt(3*a(n-1)^2 + 1). - Lekraj Beedassy, Feb 18 2002

a(n) = -a(-n) for all integer n. - Michael Somos, Sep 19 2008

Limit_{n->infinity} a(n)/a(n-1) = 2 + sqrt(3). - Gregory V. Richardson, Oct 06 2002

Binomial transform of A002605.

E.g.f.: exp(2*x)*sinh(sqrt(3)*x)/sqrt(3).

a(n) = S(n-1, 4) = U(n-1, 2); S(-1, x) := 0, Chebyshev's polynomials of the second kind A049310.

a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k, k)(-1)^k*4^(n - 2*k). - Paul Barry, Oct 25 2004

a(n) = Sum_{k=0..n-1} binomial(n+k,2*k+1)*2^k. - Paul Barry, Nov 30 2004

a(n) = 3*a(n-1) + 3*a(n-2) - a(n-3), n>=3. - Lekraj Beedassy, Jul 13 2006

a(n) = -A106707(n). - R. J. Mathar, Jul 07 2006

M^n * [1,0] = [A001075(n), A001353(n)], where M = the 2 X 2 matrix [2,3; 1,2]; e.g., a(4) = 56 since M^4 * [1,0] = [97, 56] = [A001075(4), A001353(4)]. - Gary W. Adamson, Dec 27 2006

Sequence satisfies 1 = f(a(n), a(n+1)) where f(u, v) = u^2 + v^2 - 4*u*v. - Michael Somos, Sep 19 2008

Rational recurrence: a(n) = (17*a(n-1)*a(n-2) - 4*(a(n-1)^2 + a(n-2)^2))/a(n-3) for n > 3. - Jaume Oliver Lafont, Dec 05 2009

If p[i] = Fibonacci(2i) and if A is the Hessenberg matrix of order n defined by A[i,j] = p[j-i+1], (i <= j), A[i,j] = -1, (i = j + 1), and A[i,j] = 0 otherwise, then, for n >= 1, a(n) = det A. - Milan Janjic, May 08 2010

a(n) = C_{n-1}^{(1)}(2), where C_n^{(m)}(x) is the Gegenbauer polynomial. - Eric W. Weisstein, Jul 16 2011

a(n) = -i*sin(n*arccos(2))/sqrt(3). - Eric W. Weisstein, Jul 16 2011

a(n) = sinh(n*arccosh(2))/sqrt(3). - Eric W. Weisstein, Jul 16 2011

a(n) = b such that Integral_{x=0..Pi/2} (sin(n*x))/(2-cos(x)) dx = c + b*log(2). - Francesco Daddi, Aug 02 2011

a(n) = sqrt(A098301(n)) = sqrt([A055793 / 3]), base 3 analog of A031150. - M. F. Hasler, Jan 16 2012

a(n+1) = Sum_{k=0..n} A101950(n,k)*3^k. - Philippe Deléham, Feb 10 2012

1, 4, 15, 56, 209, ... = INVERT(INVERT(1, 2, 3, 4, 5, ...)). - David Callan, Oct 13 2012

Product_{n >= 1} (1 + 1/a(n)) = 1 + sqrt(3). - Peter Bala, Dec 23 2012

Product_{n >= 2} (1 - 1/a(n)) = 1/4*(1 + sqrt(3)). - Peter Bala, Dec 23 2012

a(n+1) = (A001834(n) + A001835(n))/2. a(n+1) + a(n) = A001834(n). a(n+1) - a(n) = A001835(n). - Richard R. Forberg, Sep 04 2013

a(n) = -(-i)^(n+1)*Fibonacci(n, 4*i), i = sqrt(-1). - G. C. Greubel, Jun 06 2019

a(n)^2 - a(m)^2 = a(n+m) * a(n-m), a(n+2)*a(n-2) = 16*a(n+1)*a(n-1) - 15*a(n)^2, a(n+3)*a(n-2) = 15*a(n+2)*a(n-1) - 14*a(n+1)*a(n) for all integer n, m. - Michael Somos, Dec 12 2019

a(n) = 2^n*Sum_{k >= n} binomial(2*k,2*n-1)*(1/3)^(k+1). Cf. A102591. - Peter Bala, Nov 29 2021

a(n) = Sum_{k > 0} (-1)^((k-1)/2)*binomial(2*n, n+k)*(k|12), where (k|12) is the Kronecker symbol. - Greg Dresden, Oct 11 2022

Sum_{k=0..n} a(k) = (a(n+1) - a(n) - 1)/2. - Prabha Sivaramannair, Sep 22 2023

a(2n+1) = A001835(n+1) * A001834(n). - M. Farrokhi D. G., Oct 15 2023

Sum_{n>=1} arctan(1/(4*a(n)^2)) = Pi/12 (A019679) (Ohtskua, 2024). - Amiram Eldar, Aug 29 2024

EXAMPLE

For example, when n = 3:

****

.***

can be packed with dominoes in 4 different ways: 3 in which the top row is tiled with two horizontal dominoes and 1 in which the top row has two vertical and one horizontal domino, as shown below, so a(2) = 4.

---- ---- ---- ||--

.||| .--| .|-- .|||

G.f. = x + 4*x^2 + 15*x^3 + 56*x^4 + 209*x^5 + 780*x^6 + 2911*x^7 + 10864*x^8 + ...

MAPLE

A001353 := proc(n) option remember; if n <= 1 then n else 4*A001353(n-1)-A001353(n-2); fi; end;

A001353:=z/(1-4*z+z**2); # Simon Plouffe in his 1992 dissertation.

seq( simplify(ChebyshevU(n-1, 2)), n=0..20); # G. C. Greubel, Dec 23 2019

MATHEMATICA

a[n_] := (MatrixPower[{{1, 2}, {1, 3}}, n].{{1}, {1}})[[2, 1]]; Table[ a[n], {n, 0, 30}] (* Robert G. Wilson v, Jan 13 2005 *)

Table[GegenbauerC[n-1, 1, 2]], {n, 0, 30}] (* Zerinvary Lajos, Jul 14 2009 *)

Table[-((I Sin[n ArcCos[2]])/Sqrt[3]), {n, 0, 30}] // FunctionExpand (* Eric W. Weisstein, Jul 16 2011 *)

Table[Sinh[n ArcCosh[2]]/Sqrt[3], {n, 0, 30}] // FunctionExpand (* Eric W. Weisstein, Jul 16 2011 *)

Table[ChebyshevU[n-1, 2], {n, 0, 30}] (* Eric W. Weisstein, Jul 16 2011 *)

a[0]:=0; a[1]:=1; a[n_]:= a[n]= 4a[n-1] - a[n-2]; Table[a[n], {n, 0, 30}] (* Alonso del Arte, Jul 19 2011 *)

LinearRecurrence[{4, -1}, {0, 1}, 30] (* Sture Sjöstedt, Dec 06 2011 *)

Round@Table[Fibonacci[2n, Sqrt[2]]/Sqrt[2], {n, 0, 30}] (* Vladimir Reshetnikov, Sep 15 2016 *)

PROG

(PARI) M = [ 1, 1, 0; 1, 3, 1; 0, 1, 1]; for(i=0, 30, print1(([1, 0, 0]*M^i)[2], ", ")) \\ Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jan 25 2005

(PARI) {a(n) = real( (2 + quadgen(12))^n / quadgen(12) )}; /* Michael Somos, Sep 19 2008 */

(PARI) {a(n) = polchebyshev(n-1, 2, 2)}; /* Michael Somos, Sep 19 2008 */

(PARI) concat(0, Vec(x/(1-4*x+x^2) + O(x^30))) \\ Altug Alkan, Oct 30 2015

(Sage) [lucas_number1(n, 4, 1) for n in range(30)] # Zerinvary Lajos, Apr 22 2009

(Sage) [chebyshev_U(n-1, 2) for n in (0..20)] # G. C. Greubel, Dec 23 2019

(Haskell)

a001353 n = a001353_list !! n

a001353_list =

0 : 1 : zipWith (-) (map (4 *) $ tail a001353_list) a001353_list

-- Reinhard Zumkeller, Aug 14 2011

(GAP) a:=[0, 1];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # Muniru A Asiru, Feb 16 2018

(Magma) I:=[0, 1]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2): n in [1..30]]; // G. C. Greubel, Jun 06 2019

(Python)

a001353 = [0, 1]

for n in range(30): a001353.append(4*a001353[-1] - a001353[-2])

print(a001353) # Gennady Eremin, Feb 05 2022

CROSSREFS

Cf. A001075, A001542, A001571, A001834, A001835, A002531, A003500, A005246, A016064, A019679, A079935, A082840.

A bisection of A002530.

Cf. A125077.

A row of A116469.

Chebyshev sequence U(n, m): A000027 (m=1), this sequence (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), A077412 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33).

Cf. A323182.

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

A001834

a(0) = 1, a(1) = 5, a(n) = 4*a(n-1) - a(n-2).
(Formerly M3890 N1598)

+10
69

1, 5, 19, 71, 265, 989, 3691, 13775, 51409, 191861, 716035, 2672279, 9973081, 37220045, 138907099, 518408351, 1934726305, 7220496869, 26947261171, 100568547815, 375326930089, 1400739172541, 5227629760075, 19509779867759, 72811489710961, 271736178976085

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Sequence also gives values of x satisfying 3*y^2 - x^2 = 2, the corresponding y being given by A001835(n+1). Moreover, quadruples(p, q, r, s) satisfying p^2 + q^2 + r^2 = s^2, where p = q and r is either p+1 or p-1, are termed nearly isosceles Pythagorean and are given by p = {x + (-1)^n}/3, r = p-(-1)^n, s = y for n > 1. - Lekraj Beedassy, Jul 19 2002

a(n) = L(n,-4)*(-1)^n, where L is defined as in A108299; see also A001835 for L(n,+4). - Reinhard Zumkeller, Jun 01 2005

a(n)= A002531(1+2*n). - Anton Vrba (antonvrba(AT)yahoo.com), Feb 14 2007

361 written in base A001835(n+1) - 1 is the square of a(n). E.g., a(12) = 2672279, A001835(13) - 1 = 1542840. We have 361_(1542840) = 3*1542840 + 6*1542840 + 1 = 2672279^2. - Richard Choulet, Oct 04 2007

The lower principal convergents to 3^(1/2), beginning with 1/1, 5/3, 19/11, 71/41, comprise a strictly increasing sequence; numerators=A001834, denominators=A001835. - Clark Kimberling, Aug 27 2008

General recurrence is a(n) = (a(1) - 1)*a(n-1) - a(n-2), a(1) >= 4, lim_{n->infinity} a(n) = x*(k*x + 1)^n, k = (a(1) - 3), x = (1 + sqrt((a(1) + 1)/(a(1) - 3)))/2. Examples in OEIS: a(1) = 4 gives A002878, primes in it A121534. a(1) = 5 gives A001834, primes in it A086386. a(1) = 6 gives A030221, primes in it A299109. a(1) = 7 gives A002315, primes in it A088165. a(1) = 8 gives A033890, primes in it not in OEIS (do there exist any?). a(1) = 9 gives A057080, primes in {71, 34649, 16908641, ...}. a(1) = 10 gives A057081, primes in it {389806471, 192097408520951, ...}. - Ctibor O. Zizka, Sep 02 2008

Inverse binomial transform of A030192. - Philippe Deléham, Nov 19 2009

For positive n, a(n) equals the permanent of the (2*n) X (2*n) tridiagonal matrix with sqrt(6)'s along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011

x-values in the solution to 3x^2 + 6 = y^2 (see A082841 for the y-values). - Sture Sjöstedt, Nov 25 2011

Pisano period lengths: 1, 1, 2, 4, 3, 2, 8, 4, 6, 3, 10, 4, 12, 8, 6, 8, 18, 6, 5, 12, ... - R. J. Mathar, Aug 10 2012

The aerated sequence (b(n))_{n>=1} = [1, 0, 5, 0, 19, 0, 71, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -2, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for a connection with Chebyshev polynomials. - Peter Bala, Mar 22 2015

Yong Hao Ng has shown that for any n, a(n) is coprime with any member of A001835 and with any member of A001075. - René Gy, Feb 26 2018

From Wolfdieter Lang, Oct 15 2020: (Start)

((-1)^n)*a(n) = X(n) = (-1)^n*(S(n, 4) + S(n-1, 4)) and Y(n) = X(n-1) gives all integer solutions (modulo sign flip between X and Y) of X^2 + Y^2 + 4*X*Y = +6, for n = -oo..+oo, with Chebyshev S polynomials (see A049310), with S(-1, x) = 0, and S(-|n|, x) = - S(|n|-2, x), for |n| >= 2.

This binary indefinite quadratic form of discriminant 12, representing 6, has only this family of proper solutions (modulo sign flip), and no improper ones.

This comment is inspired by a paper by Robert K. Moniot (private communication). See his Oct 04 2020 comment in A027941 related to the case of x^2 + y^2 - 3*x*y = -1 (special Markov solutions). (End)

Floretion Algebra Multiplication Program, FAMP Code: A001834 = (4/3)vesseq[ - .25'i + 1.25'j - .25'k - .25i' + 1.25j' - .25k' + 1.25'ii' + .25'jj' - .75'kk' + .75'ij' + .25'ik' + .75'ji' - .25'jk' + .25'ki' - .25'kj' + .25e], apart from initial term

REFERENCES

Bastida, Julio R. Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163--166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009)

L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 375.

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P.-F. Teilhet, Reply to Query 2094, L'Intermédiaire des Mathématiciens, 10 (1903), 235-238.

LINKS

T. D. Noe, Table of n, a(n) for n=0..200

Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.

J. B. Cosgrave and K. Dilcher, A role for generalized Fermat numbers, Math. Comp. 86 (2017), 899-933; see also Paper #10.

Bruno Deschamps, Sur les bonnes valeurs initiales de la suite de Lucas-Lehmer, Journal of Number Theory, Volume 130, Issue 12, December 2010, Pages 2658-2670.

L. Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil.

Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.

Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 17.

Tanya Khovanova, Recursive Sequences

Seong Ju Kim, R. Stees, and L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), # 16.1.4

Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.

Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eq. (44) rhs, m=6.

Ioana-Claudia Lazăr, Lucas sequences in t-uniform simplicial complexes, arXiv:1904.06555 [math.GR], 2019.

Donatella Merlini and Renzo Sprugnoli, Arithmetic into geometric progressions through Riordan arrays, Discrete Mathematics 340.2 (2017): 160-174.

Yong Hao Ng, A partition in three classes of the set of all prime numbers?, Math StackExchange.

S. Northshield, An Analogue of Stern's Sequence for Z[sqrt(2)], Journal of Integer Sequences, 18 (2015), #15.11.6.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions and Conjectures, Universite du Quebec a Montreal, 1992.

Ryan Stees, Sequences of Spiral Knot Determinants, Senior Honors Projects, Paper 84, James Madison Univ., May 2016.

F. V. Waugh and M. W. Maxfield, Side-and-diagonal numbers, Math. Mag., 40 (1967), 74-83.

H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012) The John Selfridge Memorial Volume.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (4,-1).

FORMULA

a(n) = ((1 + sqrt(3))^(2*n + 1) + (1 - sqrt(3))^(2*n + 1))/2^(n + 1). - N. J. A. Sloane, Nov 10 2009

a(n) = (1/2) * ((1 + sqrt(3))*(2 + sqrt(3))^n + (1 - sqrt(3))*(2 - sqrt(3))^n). - Dean Hickerson, Dec 01 2002

From Mario Catalani, Apr 11 2003: (Start)

With a = 2 + sqrt(3), b = 2 - sqrt(3): a(n) = (1/sqrt(2))(a^(n + 1/2) - b^(n + 1/2)).

a(n) - a(n-1) = A003500(n).

a(n) = sqrt(1 + 12*A061278(n) + 12*A061278(n)^2). (End)

a(n) = ((1 + sqrt(3))^(2*n + 1) + (1 - sqrt(3))^(2*n + 1))/2^(n + 1). - Anton Vrba, Feb 14 2007

G.f.: (1 + x)/((1 - 4*x + x^2)). Simon Plouffe in his 1992 dissertation.

a(n) = S(2*n, sqrt(6)) = S(n, 4) + S(n-1, 4); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 4) = A001353(n).

For all members x of the sequence, 3*x^2 + 6 is a square. Limit_{n->infinity} a(n)/a(n-1) = 2 + sqrt(3). - Gregory V. Richardson, Oct 10 2002

a(n) = 2*A001571(n) + 1. - Bruce Corrigan (scentman(AT)myfamily.com), Nov 04 2002

Let q(n, x) = Sum_{i=0..n} x^(n - i)*binomial(2*n - i, i); then (-1)^n*q(n, -6) = a(n). - Benoit Cloitre, Nov 10 2002

a(n) = 2^(-n)*Sum_{k>=0} binomial(2*n + 1, 2*k)*3^k; see A091042. - Philippe Deléham, Mar 01 2004

a(n) = floor(sqrt(3)*A001835(n+1)). - Philippe Deléham, Mar 03 2004

a(n+1) - 2*a(n) = 3*A001835(n+1). Using the known relation A001835(n+1) = sqrt((a(n)^2 + 2)/3) it follows that a(n+1) - 2*a(n) = sqrt(3*(a(n)^2 + 2)). Therefore a(n+1)^2 + a(n)^2 - 4*a(n+1)*a(n) - 6 = 0. - Creighton Dement, Apr 18 2005

a(n) = Jacobi_P(n, 1/2, -1/2, 2)/Jacobi_P(n, -1/2, 1/2, 1). - Paul Barry, Feb 03 2006

Equals binomial transform of A026150 starting (1, 4, 10, 28, 76, ...) and double binomial transform of (1, 3, 3, 9, 9, 27, 27, 81, 81, ...). - Gary W. Adamson, Nov 30 2007

Sequence satisfies 6 = f(a(n), a(n+1)) where f(u, v) = u^2 + v^2 - 4*u*v. - Michael Somos, Sep 19 2008

a(-1-n) = -a(n). - Michael Somos, Sep 19 2008

From Franck Maminirina Ramaharo, Nov 11 2018: (Start)

a(n) = (-1)^n*(5*A125905(n) + A125905(n+1)).

E.g.f.: exp(2*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)). (End)

a(n) = A061278(n+1) - A061278(n-1) for n>=2. - John P. McSorley, Jun 20 2020

EXAMPLE

G.f. = 1 + 5*x + 19*x^2 + 71*x^3 + 265*x^4 + 989*x^5 + 3691*x^6 + ...

MAPLE

f:=n->((1+sqrt(3))^(2*n+1)+(1-sqrt(3))^(2*n+1))/2^(n+1); # N. J. A. Sloane, Nov 10 2009

MATHEMATICA

a[0] = 1; a[1] = 5; a[n_] := a[n] = 4a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 25}] (* Robert G. Wilson v, Apr 24 2004 *)

Table[Expand[((1+Sqrt[3])^(2*n+1)+(1+Sqrt[3])^(2*n+1))/2^(n+1)], {n, 0, 20}] (* Anton Vrba, Feb 14 2007 *)

LinearRecurrence[{4, -1}, {1, 5}, 50] (* Sture Sjöstedt, Nov 27 2011 *)

a[c_, n_] := Module[{},

p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];

d := Numerator[Convergents[Sqrt[c], n p]];

t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];

Return[t];

] (* Complement of A002531 *)

a[3, 20] (* Gerry Martens, Jun 07 2015 *)

Round@Table[LucasL[2n+1, Sqrt[2]]/Sqrt[2], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 15 2016 *)

PROG

(PARI) {a(n) = real( (2 + quadgen(12))^n * (1 + quadgen(12)) )}; /* Michael Somos, Sep 19 2008 */

(PARI) {a(n) = subst( polchebyshev(n-1, 2) + polchebyshev(n, 2), x, 2)}; /* Michael Somos, Sep 19 2008 */

(SageMath) [(lucas_number2(n, 4, 1)-lucas_number2(n-1, 4, 1))/2 for n in range(1, 27)] # Zerinvary Lajos, Nov 10 2009

(Haskell)

a001834 n = a001834_list !! (n-1)

a001834_list = 1 : 5 : zipWith (-) (map (* 4) $ tail a001834_list) a001834_list

-- Reinhard Zumkeller, Jan 23 2012

(Magma) I:=[1, 5]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 22 2015

CROSSREFS

A bisection of sequence A002531.

Cf. A001352, A001835, A086386 (prime members).

Cf. A026150.

Cf. A082841, A100047.

Cf. A002531, A001353, A049310.

a(n)^2+1 = A094347(n+1).

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

A002530

a(n) = 4*a(n-2) - a(n-4) for n > 1, a(n) = n for n = 0, 1.
(Formerly M2363 N0934)

+10
68

0, 1, 1, 3, 4, 11, 15, 41, 56, 153, 209, 571, 780, 2131, 2911, 7953, 10864, 29681, 40545, 110771, 151316, 413403, 564719, 1542841, 2107560, 5757961, 7865521, 21489003, 29354524, 80198051, 109552575, 299303201, 408855776, 1117014753, 1525870529, 4168755811

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

Denominators of continued fraction convergents to sqrt(3), for n >= 1.

Also denominators of continued fraction convergents to sqrt(3) - 1. See A048788 for numerators. - N. J. A. Sloane, Dec 17 2007. Convergents are 1, 2/3, 3/4, 8/11, 11/15, 30/41, 41/56, 112/153, ...

Consider the mapping f(a/b) = (a + 3*b)/(a + b). Taking a = b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 1/1, 2/1, 5/3, 7/4, 19/11, ... converging to 3^(1/2). Sequence contains the denominators. The same mapping for N, i.e., f(a/b) = (a + Nb)/(a + b) gives fractions converging to N^(1/2). - Amarnath Murthy, Mar 22 2003

Sqrt(3) = 2/2 + 2/3 + 2/(3*11) + 2/(11*41) + 2/(41*153) + 2/(153*571), ...; the sum of the first 6 terms of this series = 1.7320490367..., while sqrt(3) = 1.7320508075... - Gary W. Adamson, Dec 15 2007

From Clark Kimberling, Aug 27 2008: (Start)

Related convergents (numerator/denominator):

lower principal convergents: A001834/A001835

upper principal convergents: A001075/A001353

intermediate convergents: A005320/A001075

principal and intermediate convergents: A143642/A140827

lower principal and intermediate convergents: A143643/A005246. (End)

Row sums of triangle A152063 = (1, 3, 4, 11, ...). - Gary W. Adamson, Nov 26 2008

From Alois P. Heinz, Apr 13 2011: (Start)

Also number of domino tilings of the 3 X (n-1) rectangle with upper left corner removed iff n is even. For n=4 the 4 domino tilings of the 3 X 3 rectangle with upper left corner removed are:

. .___. . .___. . .___. . .___.

._|___| ._|___| ._| | | ._|___|

| |___| | | | | | |_|_| |___| |

|_|___| |_|_|_| |_|___| |___|_| (End)

This is the sequence of Lehmer numbers u_n(sqrt(R),Q) with the parameters R = 2 and Q = -1. It is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all natural numbers n and m. - Peter Bala, Apr 18 2014

2^(-floor(n/2))*(1 + sqrt(3))^n = A002531(n) + a(n)*sqrt(3); integers in the real quadratic number field Q(sqrt(3)). - Wolfdieter Lang, Feb 11 2018

Let T(n) = 2^(n mod 2), U(n) = a(n), V(n) = A002531(n), x(n) = V(n)/U(n). Then T(n*m) * U(n+m) = U(n)*V(m) + U(m)*V(n), T(n*m) * V(n+m) = 3*U(n)*U(m) + V(m)*V(n), x(n+m) = (3 + x(n)*x(m))/(x(n) + x(m)). - Michael Somos, Nov 29 2022

REFERENCES

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

Russell Lyons, A bird's-eye view of uniform spanning trees and forests, in Microsurveys in Discrete Probability, AMS, 1998.

I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 181.

Murat Sahin and Elif Tan, Conditional (strong) divisibility sequences, Fib. Q., 56 (No. 1, 2018), 18-31.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..2000

Peter Bala, Notes on 2-periodic continued fractions and Lehmer sequences

Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.

Mario Catalani, Sequences related to convergents to square root of rationals, arXiv:math/0305270 [math.NT], 2003.

Marcia Edson, Scott Lewis and Omer Yayenie, The k-periodic Fibonacci sequence and an extended Binet's formula, INTEGERS 11 (2011) #A32.

Aviezri S. Fraenkel, Jonathan Levitt, and Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.

Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.

Clark Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.

MacTutor, D'Arcy Thompson on Greek irrationals

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith [Annotated scanned copy]

D'Arcy Thompson, Excess and Defect: Or the Little More and the Little Less, Mind, New Series, Vol. 38, No. 149 (Jan., 1929), pp. 43-55 (13 pages). See page 48.

Hein van Winkel, Q-quadrangles inscribed in a circle, 2014. See Table 1. [Reference from Antreas Hatzipolakis, Jul 14 2014]

Russell Walsmith, CL-Chemy Transforms Fibonacci-Type Sequences to Arrays

E. W. Weisstein, MathWorld: Lehmer Number

Index entries for "core" sequences

Index entries for two-way infinite sequences

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

Index entries for sequences related to Chebyshev polynomials.

FORMULA

G.f.: x*(1 + x - x^2)/(1 - 4*x^2 + x^4).

a(n) = 4*a(n-2) - a(n-4). [Corrected by László Szalay, Feb 21 2014]

a(n) = -(-1)^n * a(-n) for all n in Z, would satisfy the same recurrence relation. - Michael Somos, Jun 05 2003

a(2*n) = a(2*n-1) + a(2*n-2), a(2*n+1) = 2*a(2*n) + a(2*n-1).

From Benoit Cloitre, Dec 15 2002: (Start)

a(2*n) = ((2 + sqrt(3))^n - (2 - sqrt(3))^n)/(2*sqrt(3)).

a(2*n) = A001353(n).

a(2*n-1) = ceiling((1 + 1/sqrt(3))/2*(2 + sqrt(3))^n) = ((3 + sqrt(3))^(2*n - 1) + (3 - sqrt(3))^(2*n - 1))/6^n.

a(2*n-1) = A001835(n). (End)

a(n+1) = Sum_{k=0..floor(n/2)} binomial(n - k, k) * 2^floor((n - 2*k)/2). - Paul Barry, Jul 13 2004

a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n/2) + k, floor((n - 1)/2 - k))*2^k. - Paul Barry, Jun 22 2005

G.f.: (sqrt(6) + sqrt(3))/12*Q(0), where Q(k) = 1 - a/(1 + 1/(b^(2*k) - 1 - b^(2*k)/(c + 2*a*x/(2*x - g*m^(2*k)/(1 + a/(1 - 1/(b^(2*k + 1) + 1 - b^(2*k + 1)/(h - 2*a*x/(2*x + g*m^(2*k + 1)/Q(k + 1)))))))))). - Sergei N. Gladkovskii, Jun 21 2012

a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, and a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even, where alpha = 1/2*(sqrt(2) + sqrt(6)) and beta = (1/2)*(sqrt(2) - sqrt(6)). Cf. A108412. - Peter Bala, Apr 18 2014

a(n) = (-sqrt(2)*i)^n*S(n, sqrt(2)*i)*2^(-floor(n/2)) = A002605(n)*2^(-floor(n/2)), n >= 0, with i = sqrt(-1) and S the Chebyshev polynomials (A049310). - Wolfdieter Lang, Feb 10 2018

a(n+1)*a(n+2) - a(n+3)*a(n) = (-1)^n, n >= 0. - Kai Wang, Feb 06 2020

E.g.f.: sinh(sqrt(3/2)*x)*(sinh(x/sqrt(2)) + sqrt(2)*cosh(x/sqrt(2)))/sqrt(3). - Stefano Spezia, Feb 07 2020

a(n) = ((1 + sqrt(3))^n - (1 - sqrt(3))^n)/(2*2^floor(n/2))/sqrt(3) = A002605(n)/2^floor(n/2). - Robert FERREOL, Apr 13 2023

EXAMPLE

Convergents to sqrt(3) are: 1, 2, 5/3, 7/4, 19/11, 26/15, 71/41, 97/56, 265/153, 362/209, 989/571, 1351/780, 3691/2131, ... = A002531/A002530 for n >= 1.

1 + 1/(1 + 1/(2 + 1/(1 + 1/2))) = 19/11 so a(5) = 11.

G.f. = x + x^2 + 3*x^3 + 4*x^4 + 11*x^5 + 15*x^6 + 41*x^7 + ... - Michael Somos, Mar 18 2022

MAPLE

a := proc(n) option remember; if n=0 then 0 elif n=1 then 1 elif n=2 then 1 elif n=3 then 3 else 4*a(n-2)-a(n-4) fi end; [ seq(a(i), i=0..50) ];

A002530:=-(-1-z+z**2)/(1-4*z**2+z**4); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

MATHEMATICA

Join[{0}, Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[3], n]]], {n, 1, 50}]] (* Stefan Steinerberger, Apr 01 2006 *)

Join[{0}, Denominator[Convergents[Sqrt[3], 50]]] (* or *) LinearRecurrence[ {0, 4, 0, -1}, {0, 1, 1, 3}, 50] (* Harvey P. Dale, Jan 29 2013 *)

a[ n_] := If[n<0, -(-1)^n, 1] SeriesCoefficient[ x*(1+x-x^2)/(1-4*x^2+x^4), {x, 0, Abs@n}]; (* Michael Somos, Apr 18 2019 *)

a[ n_] := ChebyshevU[n-1, Sqrt[-1/2]]*Sqrt[2]^(Mod[n, 2]-1)/I^(n-1) //Simplify; (* Michael Somos, Nov 29 2022 *)

PROG

(PARI) {a(n) = if( n<0, -(-1)^n * a(-n), contfracpnqn(vector(n, i, 1 + (i>1) * (i%2)))[2, 1])}; /* Michael Somos, Jun 05 2003 */

(PARI) { default(realprecision, 2000); for (n=0, 50, a=contfracpnqn(vector(n, i, 1+(i>1)*(i%2)))[2, 1]; write("b002530.txt", n, " ", a); ); } \\ Harry J. Smith, Jun 01 2009

(PARI) apply( {A002530(n, w=quadgen(12))=real((2+w)^(n\/2)*if(bittest(n, 0), 1-w/3, w/3))}, [0..30]) \\ M. F. Hasler, Nov 04 2019

(Magma) I:=[0, 1, 1, 3]; [n le 4 select I[n] else 4*Self(n-2) - Self(n-4): n in [1..50]]; // G. C. Greubel, Feb 25 2019

(Sage) (x*(1+x-x^2)/(1-4*x^2+x^4)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Feb 25 2019

(Python)

from functools import cache

@cache

def a(n): return [0, 1, 1, 3][n] if n < 4 else 4*a(n-2) - a(n-4)

print([a(n) for n in range(36)]) # Michael S. Branicky, Nov 13 2022

CROSSREFS

Cf. A002531 (numerators of convergents to sqrt(3)), A048788, A003297.

Bisections: A001353 and A001835.

Cf. A152063.

Cf. A108412, A049310.

Analog for sqrt(m): A000129 (m=2), A001076 (m=5), A041007 (m=6), A041009 (m=7), A041011 (m=8), A005668 (m=10), A041015 (m=11), A041017 (m=12), ..., A042935 (m=999), A042937 (m=1000).

KEYWORD

nonn,easy,frac,core,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Definition edited by M. F. Hasler, Nov 04 2019

STATUS

approved

A080040

a(n) = 2*a(n-1) + 2*a(n-2) for n > 1; a(0)=2, a(1)=2.

+10
35

2, 2, 8, 20, 56, 152, 416, 1136, 3104, 8480, 23168, 63296, 172928, 472448, 1290752, 3526400, 9634304, 26321408, 71911424, 196465664, 536754176, 1466439680, 4006387712, 10945654784, 29904084992, 81699479552, 223207129088, 609813217280, 1666040692736, 4551707820032

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

The Lucas sequence V_n(2,-2). - Jud McCranie, Mar 02 2012

The signed version 2, -2, 8, -20, 56, -152, 416, -1136, 3104, -8480, 23168, ... is the Lucas sequence V(-2,-2). - R. J. Mathar, Jan 08 2013

After a(2) equals round((1+sqrt(3))^n) = 1, 3, 7, 20, 56, 152, ... - Jeremy Gardiner, Aug 11 2013

Also the number of independent vertex sets and vertex covers in the n-sunlet graph. - Eric W. Weisstein, Sep 27 2017

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, and M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239v1 [math.CO], 2015-2017.

Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

D. Jhala, G. P. S. Rathore, and K. Sisodiya, Some Properties of k-Jacobsthal Numbers with Arithmetic Indexes, Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 4, 119-124.

Tanya Khovanova, Recursive Sequences

D. H. Lehmer, On Lucas's test for the primality of Mersenne's numbers, Journal of the London Mathematical Society 1.3 (1935): 162-165. See V_n.

Eric Weisstein's World of Mathematics, Independent Vertex Set

Eric Weisstein's World of Mathematics, Sunlet Graph

Eric Weisstein's World of Mathematics, Vertex Cover

Index entries for linear recurrences with constant coefficients, signature (2,2).

Index entries for Lucas sequences (2,-2).

FORMULA

G.f.: (2-2*x)/(1-2*x-2*x^2).

a(n) = (1+sqrt(3))^n + (1-sqrt(3))^n.

a(n) = 2*A026150(n). - Philippe Deléham, Nov 19 2008

G.f.: G(0), where G(k) = 1 + 1/(1 - x*(3*k-1)/(x*(3*k+2) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 11 2013

a(n) = 2*2^floor(n/2)*A002531(n). - Ralf Stephan, Sep 08 2013

a(n) = [x^n] ( 1 + x + sqrt(1 + 2*x + 3*x^2) )^n for n >= 1. - Peter Bala, Jun 29 2015

E.g.f.: 2*exp(x)*cosh(sqrt(3)*x). - Stefano Spezia, Mar 02 2024

MATHEMATICA

CoefficientList[Series[(2 - 2 t)/(1 - 2 t - 2 t^2), {t, 0, 30}], t]

With[{c = {2, 2}}, LinearRecurrence[c, c, 20]] (* Harvey P. Dale, Apr 24 2016 *)

Round @ Table[LucasL[n, Sqrt[2]] 2^(n/2), {n, 0, 20}] (* Vladimir Reshetnikov, Sep 15 2016 *)

Table[(1 - Sqrt[3])^n + (1 + Sqrt[3])^n, {n, 0, 20}] // Expand (* Eric W. Weisstein, Sep 27 2017 *)

PROG

(Sage) from sage.combinat.sloane_functions import recur_gen2b; it = recur_gen2b(2, 2, 2, 2, lambda n: 0); [next(it) for i in range(27)] # Zerinvary Lajos, Jul 16 2008

(Sage) [lucas_number2(n, 2, -2) for n in range(0, 27)] # Zerinvary Lajos, Apr 30 2009

(Haskell)

a080040 n = a080040_list !! n

a080040_list =

2 : 2 : map (* 2) (zipWith (+) a080040_list (tail a080040_list))

-- Reinhard Zumkeller, Oct 15 2011

(PARI) a(n)=([0, 1; 2, 2]^n*[2; 2])[1, 1] \\ Charles R Greathouse IV, Apr 08 2016

(Magma) a:=[2, 2]; [n le 2 select a[n] else 2*Self(n-1) + 2*Self(n-2):n in [1..27]]; Marius A. Burtea, Jan 20 2020

(Magma) R<x>:=PowerSeriesRing(Rationals(), 27); Coefficients(R!( (2-2*x)/(1-2*x-2*x^2))); // Marius A. Burtea, Jan 20 2020

CROSSREFS

Cf. A002531, A002605, A028859, A030195, A083337, A106435, A108898, A125145.

Equals 2*A026150.

KEYWORD

easy,nonn

AUTHOR

Mario Catalani (mario.catalani(AT)unito.it), Jan 21 2003

STATUS

approved

A002812

a(n) = 2*a(n-1)^2 - 1, starting a(0) = 2.
(Formerly M1817 N0720)

+10
18

2, 7, 97, 18817, 708158977, 1002978273411373057, 2011930833870518011412817828051050497, 8095731360557835890888779535060256832479295062749579257164654370487894017

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

An infinite coprime sequence defined by recursion. - Michael Somos, Mar 14 2004

2^p-1 is prime iff it divides a(p-2), since a(n) = A003010(n)/2, where A003010 is the Lucas-Lehmer sequence used for Mersenne number primality testing. - M. F. Hasler, Mar 09 2007

From Cino Hilliard, Sep 28 2008: (Start)

Also numerators of the convergents to the square root of 3 using the following recursion for initial x = 1: x1=x, x=3/x, x=(x+x1)/2.

This recursion was derived by experimenting with polynomial recursions of the form x = -a(0)/(a(n-1)x^(n-1) + ... + a(1)) in an effort to find a root for the polynomial a(n)x^n + a(n-1)x^(n-1) + ... + a(0). The process was hit-and-miss until I introduced the averaging step described above. This method is equivalent to Newton's Method although derived somewhat differently. (End)

The sequence satisfies the Pell equation a(n)^2 - 3*A071579(n)^2 = 1. - Vincenzo Librandi, Dec 19 2011

From Peter Bala, Nov 2012: (Start)

The present sequence corresponds to the case x = 2 of the following general remarks. Let x > 1 and let alpha := {x + sqrt(x^2 - 1)}. Define a sequence a(n) (which depends on x) by setting a(n) = (1/2)*(alpha^(2^n) + (1/alpha)^(2^n)) for n >= 0. It is easy to verify that a(n) is a solution to the recurrence equation a(n+1) = 2*a(n)^2 - 1 with the initial condition a(0) = x.

The following algebraic identity is valid for x > 1:

sqrt(4*x^2 - 4)/(2*x + 1) = (1 - 1/(2*x))*sqrt(4*y^2 - 4)/(2*y + 1), where y = 2*x^2 - 1. Iterating the identity yields the product expansion sqrt(4*x^2 - 4)/(2*x + 1) = Product_{n >= 0} (1 - 1/(2*a(n))). A second expansion is Product_{n >= 0} (1 + 1/a(n)) = sqrt((x + 1)/(x - 1)). See Mendes-France and van der Poorten. (End)

REFERENCES

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 399.

E. Lucas, Nouveaux théorèmes d'arithmétique supérieure, Comptes Rend., 83 (1876), 1286-1288.

W. Sierpiński, Sur les développements systématiques des nombres en produits infinis, in Œuvres choisies, tome 1, PWN Editions scientifiques de Pologne, 1974.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10

Georg Cantor, Zwei Sätze über eine gewisse Zerlegung der Zahlen in unendliche Producte, Zeitschrift für Mathematik und Physik. Band 14, 1869, S. 152-158. (See page 155, II, error in the fourth term.)

E. Lucas, Nouveaux théorèmes d'arithmétique supérieure (annotated scanned copy)

M. Mendes France and A. J. van der Poorten, From geometry to Euler identities, Theoret. Comput. Sci., 65 (1989), 213-220.

Jeffrey Shallit, Rational numbers with non-terminating, non-periodic modified Engel-type expansions, Fib. Quart., 31 (1993), 37-40.

Eric Weisstein's World of Mathematics, Newton's Iteration

Index entries for sequences related to Engel expansions

FORMULA

a(n) = A001075(2^n).

a(n) = ((2+sqrt(3))^(2^n) + (2-sqrt(3))^(2^n))/2. - Bruno Berselli, Dec 20 2011

From Peter Bala, Nov 11 2012: (Start)

2*sqrt(3)/5 = Product_{n >= 0} (1 - 1/(2*a(n))).

sqrt(3) = Product_{n >= 0} (1 + 1/a(n)).

a(n) = (1/2)*A003010(n). (End)

a(n) = cos(2^n*arccos(2)). - Peter Luschny, Oct 12 2022

a(n) = A002531(2^(n+1)). - Robert FERREOL, Apr 13 2023

EXAMPLE

G.f. = 2 + 7*x + 97*x^2 + 18817*x^3 + 708158977*x^4 + ...

MAPLE

a:=n->((2+sqrt(3))^(2^n)+(2-sqrt(3))^(2^n))/2: seq(floor(a(n)), n=0..10); # Muniru A Asiru, Aug 12 2018

MATHEMATICA

Table[((2 + Sqrt[3])^2^n + (2 - Sqrt[3])^2^n)/2, {n, 0, 7}] (* Bruno Berselli, Dec 20 2011 *)

NestList[2#^2-1&, 2, 10] (* Harvey P. Dale, May 04 2013 *)

a[ n_] := If[ n < 0, 0, ChebyshevT[2^n, 2]]; (* Michael Somos, Dec 06 2016 *)

PROG

(PARI) {a(n) = if( n<1, 2 * (n==0), 2 * a(n-1)^2 - 1)}; /* Michael Somos, Mar 14 2004 */

(PARI) /* Roots by recursion. Find first root of ax^2 + b^x + c */

rroot2(a, b, c, p) = { local(x=1, x1=1, j); for(j=1, p, x1=x; x=-c/(a*x+b); x=(x1+x)/2; /* Let x be the average of the last 2 values */ print1(numerator(x)", "); ); } \\ Cino Hilliard, Sep 28 2008

(Magma) I:=[2]; [n le 1 select I[n] else 2*Self(n-1)^2-1: n in [1..8]]; // Vincenzo Librandi, Dec 19 2011

(GAP) a:=[2];; for n in [2..10] do a[n]:=2*a[n-1]^2-1; od; a; # Muniru A Asiru, Aug 12 2018

CROSSREFS

Cf. A001075, A002531, A003010, A071579.

Cf. A177879 (lpf).

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

A041006

Numerators of continued fraction convergents to sqrt(6).

+10
15

2, 5, 22, 49, 218, 485, 2158, 4801, 21362, 47525, 211462, 470449, 2093258, 4656965, 20721118, 46099201, 205117922, 456335045, 2030458102, 4517251249, 20099463098, 44716177445, 198964172878, 442644523201, 1969542265682, 4381729054565, 19496458483942

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Interspersion of 2 sequences, 2*A054320 and A001079. - Gerry Martens, Jun 10 2015

LINKS

Hugo Pfoertner, Table of n, a(n) for n = 0..100

Index entries for linear recurrences with constant coefficients, signature (0,10,0,-1).

FORMULA

From M. F. Hasler, Feb 13 2009: (Start)

a(2n) = 2*A142238(2n) = A041038(2n)/2;

a(2n-1) = A142238(2n-1) = A041038(2n-1) = A001079(n). (End)

G.f.: (2 + 5*x + 2*x^2 - x^3)/(1 - 10*x^2 + x^4).

MATHEMATICA

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[6], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011*)

LinearRecurrence[{0, 10, 0, -1}, {2, 5, 22, 49}, 50] (* Vincenzo Librandi, Jun 10 2015 *)

PROG

(Magma) I:=[2, 5, 22, 49]; [n le 4 select I[n] else 10*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 10 2015

From M. F. Hasler, Nov 01 2019: (Start)

(PARI) A41006=contfracpnqn(c=contfrac(sqrt(6)), #c)[1, ][^-1] \\ Discard possibly incorrect last element. NB: a(n)=A41006[n+1]! For correct index & more terms:

A041006(n)={n<#A041006|| A041006=extend(A041006, [2, 10; 4, -1], n\.8); A041006[n+1]}

extend(A, c, N)={for(n=#A+1, #A=Vec(A, N), A[n]=[A[n-i]|i<-c[, 1]]*c[, 2]); A} \\ (End)

CROSSREFS

Cf. A041007 (denominators).

Cf. A054320, A001079.

Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041008 (m=7), A041010 (m=8), A005667 (m=10), A041014 (m=11), ..., A042936 (m=1000).

KEYWORD

nonn,cofr,frac,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vincenzo Librandi, Jun 10 2015

STATUS

approved

A048788

a(2n+1) = a(2n) + a(2n-1), a(2n) = 2*a(2n-1) + a(2n-2); a(n) = n for n = 0, 1.

+10
13

0, 1, 2, 3, 8, 11, 30, 41, 112, 153, 418, 571, 1560, 2131, 5822, 7953, 21728, 29681, 81090, 110771, 302632, 413403, 1129438, 1542841, 4215120, 5757961, 15731042, 21489003, 58709048, 80198051, 219105150, 299303201, 817711552, 1117014753

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Numerators of continued fraction convergents to sqrt(3) - 1 (A160390). See A002530 for denominators. - N. J. A. Sloane, Dec 17 2007

Convergents are 1, 2/3, 3/4, 8/11, 11/15, 30/41, 41/56, 112/153, ... - Clark Kimberling, Sep 21 2013

A strong divisibility sequence, that is gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. - Peter Bala, Jun 06 2014

From Sarah-Marie Belcastro, Feb 15 2022: (Start)

a(n) is also the number of perfect matchings of an edge-labeled 2 X (n-1) Mobius band grid graph, or equivalently the number of domino tilings of a 2 X (n-1) Mobius band grid. (The twist is on the length-n side.)

a(n) is also the output of Lu and Wu's formula for the number of perfect matchings of an m X n Mobius band grid, specialized to m = 2 with the twist on the length-n side.

2*a(n) is the number of perfect matchings of an edge-labeled 2 X (n-1) projective planar grid graph, or equivalently the number of domino tilings of a 2 X (n-1) projective planar grid. (End)

REFERENCES

Russell Lyons, A bird's-eye view of uniform spanning trees and forests, in Microsurveys in Discrete Probability, AMS, 1998.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Peter Bala, Notes on 2-periodic continued fractions and Lehmer sequences

Sarah-Marie Belcastro, Domino Tilings of 2 X n Grids (or Perfect Matchings of Grid Graphs) on Surfaces, J. Integer Seq. 26 (2023), Article 23.5.6.

Marcia Edson, Scott Lewis and Omer Yayenie, The k-periodic Fibonacci sequence and an extended Binet's formula, INTEGERS 11 (2011) #A32.

W. T. Lu and F. Y. Wu, Close-packed dimers on nonorientable surfaces, Physics Letters A, 293 (2002), 235-246.

D. Panario, M. Sahin, and Q. Wang, A family of Fibonacci-like conditional sequences, INTEGERS, Vol. 13, 2013, #A78.

J. L. Ramirez and F. Sirvent, A q-Analogue of the Bi-Periodic Fibonacci Sequence, J. Int. Seq. 19 (2016) # 16.4.6, t_n at a=2, b=1.

Index to divisibility sequences

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

FORMULA

G.f.: x*(1+2*x-x^2)/(1-4*x^2+x^4). - Paul Barry, Sep 18 2009

a(n) = 4*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 10 2013

a(2*n-1) = A001835(n); a(2*n) = 2*A001353(n). - Peter Bala, Jun 06 2014

From Gerry Martens, Jul 11 2015: (Start)

Interspersion of 2 sequences [a1(n-1),a0(n)] for n>0:

a0(n) = ((3+sqrt(3))*(2-sqrt(3))^n-((-3+sqrt(3))*(2+sqrt(3))^n))/6.

a1(n) = 2*Sum_{i=1..n} a0(i). (End)

a(n) = ((r + (-1)^n/r)*s^n/2^(n/2) - (1/r + (-1)^n*r)*2^(n/2)/s^n)*sqrt(6)/12, where r = 1 + sqrt(2), s = 1 + sqrt(3). - Vladimir Reshetnikov, May 11 2016

a(n) = 2*ChebyshevU(n-1, 2) if n is even and ChebyshevU(n, 2) - ChebyshevU(n-1, 2) if n in odd. - G. C. Greubel, Dec 23 2019

a(n) = -(-1)^n*a(-n) for all n in Z. - Michael Somos, Sep 17 2020

MAPLE

seq( simplify( `if`(`mod`(n, 2)=0, 2*ChebyshevU((n-2)/2, 2), ChebyshevU((n-1)/2, 2) - ChebyshevU((n-3)/2, 2)) ), n=0..40); # G. C. Greubel, Dec 23 2019

MATHEMATICA

Numerator[NestList[(2/(2 + #))&, 0, 40]] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2010 *)

CoefficientList[Series[x(1+2x-x^2)/(1-4x^2+x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 10 2013 *)

a0[n_]:= ((3+Sqrt[3])*(2-Sqrt[3])^n-((-3+Sqrt[3])*(2+Sqrt[3])^n))/6 // Simplify

a1[n_]:= 2*Sum[a0[i], {i, 1, n}]

Flatten[MapIndexed[{a1[#-1], a0[#]}&, Range[20]]] (* Gerry Martens, Jul 10 2015 *)

Round@Table[With[{r=1+Sqrt[2], s=1+Sqrt[3]}, ((r + (-1)^n/r) s^n/2^(n/2) - (1/r + (-1)^n r) 2^(n/2)/s^n) Sqrt[6]/12], {n, 0, 20}] (* or *) LinearRecurrence[ {0, 4, 0, -1}, {0, 1, 2, 3}, 40] (* Vladimir Reshetnikov, May 11 2016 *)

Table[If[EvenQ[n], 2*ChebyshevU[(n-2)/2, 2], ChebyshevU[(n-1)/2, 2] - ChebyshevU[(n-3)/2, 2]], {n, 0, 40}] (* G. C. Greubel, Dec 23 2019 *)

PROG

(Magma) I:=[0, 1, 2, 3]; [n le 4 select I[n] else 4*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 10 2013

(PARI) main(size)=v=vector(size); v[1]=0; v[2]=1; v[3]=2; v[4]=3; for(i=5, size, v[i]=4*v[i-2] - v[i-4]); v; \\ Anders Hellström, Jul 11 2015

(PARI) a=vector(50); a[1]=1; a[2]=2; for(n=3, #a, if(n%2==1, a[n]=a[n-1]+a[n-2], a[n]=2*a[n-1]+a[n-2])); concat(0, a) \\ Colin Barker, Jan 30 2016

(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -1, 0, 4, 0]^n*[0; 1; 2; 3])[1, 1] \\ Charles R Greathouse IV, Mar 16 2017

(PARI) apply( {A048788(n)=imag((2+quadgen(12))^(n\/2)*if(bittest(n, 0), quadgen(12)-1, 2))}, [0..30]) \\ M. F. Hasler, Nov 04 2019

(PARI) {a(n) = my(s=1, m=n); if(n<0, s=-(-1)^n; m=-n); polcoeff(x*(1+2*x-x^2)/(1-4*x^2+x^4) + x*O(x^m), m)*s}; /* Michael Somos, Sep 17 2020 */

(Sage)

@CachedFunction

def a(n):

if (mod(n, 2)==0): return 2*chebyshev_U((n-2)/2, 2)

else: return chebyshev_U((n-1)/2, 2) - chebyshev_U((n-3)/2, 2)

[a(n) for n in (0..40)] # G. C. Greubel, Dec 23 2019

(GAP) a:=[0, 1, 2, 3];; for n in [5..40] do a[n]:=4a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 23 2019

CROSSREFS

Cf. A002530, A002531.

Bisections are A001835 and A052530.

KEYWORD

nonn,easy,frac

AUTHOR

Robin Trew (trew(AT)hcs.harvard.edu), Dec 11 1999

EXTENSIONS

Denominator of g.f. corrected by Paul Barry, Sep 18 2009

Incorrect g.f. deleted by Colin Barker, Aug 10 2012

STATUS

approved

A041008

Numerators of continued fraction convergents to sqrt(7).

+10
11

2, 3, 5, 8, 37, 45, 82, 127, 590, 717, 1307, 2024, 9403, 11427, 20830, 32257, 149858, 182115, 331973, 514088, 2388325, 2902413, 5290738, 8193151, 38063342, 46256493, 84319835, 130576328, 606625147, 737201475, 1343826622, 2081028097, 9667939010, 11748967107, 21416906117

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

C. Elsner, Series of Error Terms for Rational Approximations of Irrational Numbers , J. Int. Seq. 14 (2011) # 11.1.4.

C. Elsner, M. Stein, On Error Sum Functions Formed by Convergents of Real Numbers, J. Int. Seq. 14 (2011) # 11.8.6.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,16,0,0,0,-1).

FORMULA

G.f.: (2 + 3*x + 5*x^2 + 8*x^3 + 5*x^4 - 3*x^5 + 2*x^6 - x^7)/(1 - 16*x^4 + x^8).

MATHEMATICA

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[7], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)

Numerator[Convergents[Sqrt[7], 30]] (* Vincenzo Librandi, Oct 28 2013 *)

LinearRecurrence[{0, 0, 0, 16, 0, 0, 0, -1}, {2, 3, 5, 8, 37, 45, 82, 127}, 40] (* Harvey P. Dale, Jul 23 2021 *)

PROG

From M. F. Hasler, Nov 01 2019: (Start)

(PARI) A041008=contfracpnqn(c=contfrac(sqrt(7)), #c)[1, ][^-1] \\ Discard possibly incorrect last element. NB: a(n)=A041008[n+1]! For more terms use:

A041008(n)={n<#A041008|| A041008=extend(A041008, [4, 16; 8, -1], n\.8); A041008[n+1]}

extend(A, c, N)={for(n=#A+1, #A=Vec(A, N), A[n]=[A[n-i]|i<-c[, 1]]*c[, 2]); A} \\ (End)

CROSSREFS

Cf. A010465, A041009 (denominators), A266698 (quadrisection), A001081 (quadrisection).

Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041010 (m=8), A005667 (m=10), A041014 (m=11), A041016 (m=12), ..., A042934 (m=999), A042936 (m=1000).

KEYWORD

nonn,cofr,frac,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A041014

Numerators of continued fraction convergents to sqrt(11).

+10
9

3, 10, 63, 199, 1257, 3970, 25077, 79201, 500283, 1580050, 9980583, 31521799, 199111377, 628855930, 3972246957, 12545596801, 79245827763, 250283080090, 1580944308303, 4993116004999, 31539640338297

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (0,20,0,-1).

FORMULA

G.f.: (3 + 10*x + 3*x^2 - x^3)/(1 - 20*x^2 + x^4).

MATHEMATICA

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[11], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)

Numerator[Convergents[Sqrt[11], 30]] (* Vincenzo Librandi, Oct 28 2013 *)

PROG

(PARI) A041014=contfracpnqn(c=contfrac(sqrt(11)), #c)[1, ][^-1] \\ Discard last element which may be incorrect. Use e.g. \p999 to get more terms, or extend as follows:

{A041014_upto(N, A=Vec(A041014, N))=for(n=#A041014+1, N, A[n]=20*A[n-2]-A[n-4]); A041014=A} \\ M. F. Hasler, Nov 01 2019

CROSSREFS

Cf. A010468, A041015 (denominators).

Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041008 (m=7), A041010 (m=8), A005667 (m=10), A041016 (m=12), ..., A042936 (m=1000).

KEYWORD

nonn,cofr,frac,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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