A041014 -id:A041014

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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

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

A042936

Numerators of continued fraction convergents to sqrt(1000).

+10
8

31, 32, 63, 95, 158, 253, 1676, 3605, 8886, 136895, 282676, 702247, 4496158, 5198405, 9694563, 14892968, 24587531, 39480499, 2472378469, 2511858968, 4984237437, 7496096405, 12480333842, 19976430247, 132338915324, 284654260895, 701647437114, 10809365817605, 22320379072324

(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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 78960998, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).

MATHEMATICA

Numerator[Convergents[Sqrt[1000], 30]] (* Harvey P. Dale, Oct 29 2013 *)

PROG

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

(PARI) A42936=contfracpnqn(c=contfrac(sqrt(1000)), #c)[1, ][^-1] \\ Discards possibly incorrect last term. NB: a(n)=A42936[n+1]. Could be extended using: {A42936=concat(A42936, 78960998*A42936[-18..-1]-A42936[-36..-19])}

\\ But terms with arbitrarily large indices can be computed using:

A042936(n)={[A42936[n%18+i]|i<-[1, 19]]*([0, -1; 1, 78960998]^(n\18))[, 1]} \\ Faster but longer with n=divrem(n, 18). (End)

CROSSREFS

Cf. A042937 (denominators).

Analog for sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041008 (m=7), A041010 (m=8), A005667 (m=10), A041014 (m=11), ..., A042934 (m=999).

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A041010

Numerators of continued fraction convergents to sqrt(8).

+10
7

2, 3, 14, 17, 82, 99, 478, 577, 2786, 3363, 16238, 19601, 94642, 114243, 551614, 665857, 3215042, 3880899, 18738638, 22619537, 109216786, 131836323, 636562078, 768398401, 3710155682, 4478554083, 21624372014, 26102926097, 126036076402, 152139002499

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

OFFSET

0,1

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..199 [1 removed by Georg Fischer, Jul 01 2019]

Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

FORMULA

a(n) = 6*a(n-2) - a(n-4).

a(2n) = a(2n-1) + a(2n-2), a(2n+1) = 4*a(2n) + a(2n-1).

a(2n) = A001333(2n), a(2n+1) = 2*A001333(2n+1).

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

a(n) = A001333(n+1)*A000034(n+1). - R. J. Mathar, Jul 08 2009

From Gerry Martens, Jul 11 2015: (Start)

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

a0(n) = -((3-2*sqrt(2))^n*(1+sqrt(2))) + (-1+sqrt(2))*(3+2*sqrt(2))^n.

a1(n) = ((3-2*sqrt(2))^n + (3+2*sqrt(2))^n)/2. (End)

MATHEMATICA

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[8], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011*)

CoefficientList[Series[(2 + 3*x + 2*x^2 - x^3)/(1 - 6*x^2 + x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 28 2013 *)

a0[n_] := -((3-2*Sqrt[2])^n*(1+Sqrt[2]))+(-1+Sqrt[2])*(3+2*Sqrt[2])^n // Simplify

a1[n_] := ((3-2*Sqrt[2])^n+(3+2*Sqrt[2])^n)/2 // Simplify

Flatten[MapIndexed[{a0[#], a1[#]} &, Range[20]]] (* Gerry Martens, Jul 11 2015 *)

PROG

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

(PARI) A041010=contfracpnqn(c=contfrac(sqrt(8)), #c)[1, ][^-1] \\ Discard possibly incorrect last element. NB: a(n)=A041010[n+1]! For more terms use:

A041010(n)={n<#A041010|| A041010=extend(A041010, [-1, 0, 6, 0]~, n\.8); A041010[n+1]}

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

CROSSREFS

Cf. A040005 (continued fraction), A041011 (denominators), A010466 (decimals).

Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041008 (m=7), 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

EXTENSIONS

Entry improved by Michael Somos

Initial term 1 removed and b-file, program and formulas adapted by Georg Fischer, Jul 01 2019

Cross-references added by M. F. Hasler, Nov 02 2019

STATUS

approved

A041015

Denominators of continued fraction convergents to sqrt(11).

+10
7

1, 3, 19, 60, 379, 1197, 7561, 23880, 150841, 476403, 3009259, 9504180, 60034339, 189607197, 1197677521, 3782639760, 23893516081, 75463188003, 476672644099, 1505481120300, 9509559365899, 30034159217997

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

OFFSET

0,2

COMMENTS

Sqrt(11) = 3 + continued fraction [3, 6, 3, 6, 3, 6, ...] = 6/2 + 6/19 + 6/(19*379) + 6/(379*7561) + ... - Gary W. Adamson, Dec 21 2007

Let X = the 2 X 2 matrix [1, 6; 3, 19], then X^n * [1, 0] = [a(n+1), a(n+2)]; e.g., X^3 * [1, 0] = [379, 1197] = [a(4), a(5)]. - Gary W. Adamson, Dec 21 2007

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.: (1+3*x-x^2)/(1-20*x^2+x^4). - Colin Barker, Dec 31 2011

From Gerry Martens, Jul 11 2015: (Start)

Interspersion of 2 sequences [a0(n),a1(n)]:

a0(n) = ((11+3*sqrt(11))/(10+3*sqrt(11))^n + (11-3*sqrt(11))*(10+3*sqrt(11))^n)/22.

a1(n) = 3*Sum_{i=1..n} a0(i). (End)

MATHEMATICA

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

a0[n_] := (11+3*Sqrt[11]+(11-3*Sqrt[11])*(10+3*Sqrt[11])^(2*n))/(22*(10+3*Sqrt[11])^n) // Simplify

a1[n_] := 3*Sum[a0[i], {i, 1, n}]

Flatten[MapIndexed[{a0[#], a1[#]}&, Range[11]]] (* Gerry Martens, Jul 10 2015 *)

CROSSREFS

Cf. A010468, A041014.

KEYWORD

nonn,cofr,frac,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A042934

Numerators of continued fraction convergents to sqrt(999).

+10
5

31, 32, 63, 95, 158, 885, 5468, 6353, 37233, 80819, 441328, 522147, 3574210, 18393197, 21967407, 40360604, 62328011, 102688615, 6429022141, 6531710756, 12960732897, 19492443653, 32453176550

(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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 205377230, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).

FORMULA

a(n) = 205377230*a(n-18) - a(n-36). - Wesley Ivan Hurt, May 28 2021

MATHEMATICA

Numerator[Convergents[Sqrt[999], 30]] (* Vincenzo Librandi, Dec 10 2013 *)

PROG

(PARI) A42934=contfracpnqn(c=contfrac(sqrt(999)), #c)[1, ][^-1] \\ Discard possibly incorrect last element. NB: a(n) = A42934[n+1]! For more terms, use:

A042934(n)={n<#A42934 || A42934_upto(n+10); A42934[n+1]}

{A42934_upto(N, A=Vec(A42934, N))=for(n=#A42934+1, N, A[n]=205377230*A[n-18]-A[n-36]); A42934=A} \\ M. F. Hasler, Nov 01 2019

CROSSREFS

Cf. A042935 (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), A041014 (m=11), ..., A042936 (m=1000).

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A123482

Coefficients of the series giving the best rational approximations to sqrt(11).

+10
5

60, 23940, 9528120, 3792167880, 1509273288180, 600686976527820, 239071907384784240, 95150018452167599760, 37869468272055319920300, 15071953222259565160679700, 5998599512991034878630600360, 2387427534217209622129818263640, 950190160018936438572789038328420

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

OFFSET

1,1

COMMENTS

The partial sums of the series 10/3 - 1/a(1) - 1/a(2) - 1/a(3) - ... give the best rational approximations to sqrt(11), which constitute every second convergent of the continued fraction. The corresponding continued fractions are [3;3,6,3], [3;3,6,3,6,3], [3;3,6,3,6,3,6,3] and so forth.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..380

Index entries for linear recurrences with constant coefficients, signature (399,-399,1).

FORMULA

a(n+3) = 399*a(n+2) - 399*a(n+1) + a(n).

a(n) = -5/33 + (5/66 + 1/44*11^(1/2))*(199 + 60*11^(1/2))^n + (5/66 - 1/44*11^(1/2))*(199 - 60*11^(1/2))^n.

G.f.: -60*x / ((x-1)*(x^2-398*x+1)). - Colin Barker, Jun 23 2014

MATHEMATICA

CoefficientList[Series[-60*x/((x - 1)*(x^2 - 398*x + 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 13 2017 *)

PROG

(PARI) Vec(-60*x/((x-1)*(x^2-398*x+1)) + O(x^100)) \\ Colin Barker, Jun 23 2014

CROSSREFS

Cf. A010468, A041014, A041015.

Cf. A123478, A123479, A123480, A029549.

KEYWORD

nonn,easy

AUTHOR

Gene Ward Smith, Oct 02 2006

EXTENSIONS

More terms from Colin Barker, Jun 23 2014

STATUS

approved

A180029

Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 + 2*x)/(1 - 6*x - 2*x^2).

+10
3

1, 8, 50, 316, 1996, 12608, 79640, 503056, 3177616, 20071808, 126786080, 800860096, 5058732736, 31954116608, 201842165120, 1274961223936, 8053451673856, 50870632491008, 321330698293760, 2029725454744576

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

OFFSET

0,2

COMMENTS

The a(n) represent the number of n-move routes of a fairy chess piece starting in the central square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a white queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen, see A180028.

The sequence above corresponds to 8 red queen vectors, i.e., A[5] vector, with decimal values 255, 383, 447, 479, 503, 507, 509 and 510. The other squares lead for these vectors to A135030.

LINKS

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

Index entries for linear recurrences with constant coefficients, signature (6, 2).

FORMULA

G.f.: (1+2*x)/(1 - 6*x - 2*x^2).

a(n) = 6*a(n-1) + 2*a(n-2) with a(0) = 1 and a(1) = 8.

a(n) = ((5-4*A)*A^(-n-1) + (5-4*B)*B^(-n-1))/22 with A = (-3+sqrt(11))/2 and B = (-3-sqrt(11))/2.

Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n-1)*A016116(n+1)/(A041015(n-1)*sqrt(11) - A041014(n-1)) for n >= 1.

MAPLE

with(LinearAlgebra): nmax:=19; m:=5; A[5]:= [0, 1, 1, 1, 1, 1, 1, 1, 1]: A:=Matrix([[0, 1, 1, 1, 1, 0, 1, 0, 1], [1, 0, 1, 1, 1, 1, 0, 1, 0], [1, 1, 0, 0, 1, 1, 1, 0, 1], [1, 1, 0, 0, 1, 1, 1, 1, 0], A[5], [0, 1, 1, 1, 1, 0, 0, 1, 1], [1, 0, 1, 1, 1, 0, 0, 1, 1], [0, 1, 0, 1, 1, 1, 1, 0, 1], [1, 0, 1, 0, 1, 1, 1, 1, 0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m, k], k=1..9): od: seq(a(n), n=0..nmax);

MATHEMATICA

LinearRecurrence[{6, 2}, {1, 8}, 50 ] (* Vincenzo Librandi, Nov 15 2011 *)

PROG

(Magma) I:=[1, 8]; [n le 2 select I[n] else 6*Self(n-1)+2*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 15 2011

KEYWORD

easy,nonn

AUTHOR

Johannes W. Meijer, Aug 09 2010

STATUS

approved

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