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A024206
Expansion of x^2*(1+x-x^2)/((1-x^2)*(1-x)^2).
46
0, 1, 3, 5, 8, 11, 15, 19, 24, 29, 35, 41, 48, 55, 63, 71, 80, 89, 99, 109, 120, 131, 143, 155, 168, 181, 195, 209, 224, 239, 255, 271, 288, 305, 323, 341, 360, 379, 399, 419, 440, 461, 483, 505, 528, 551, 575, 599, 624, 649, 675, 701, 728, 755, 783, 811, 840
OFFSET
1,3
COMMENTS
a(n+1) is the number of 2 X n binary matrices with no zero rows or columns, up to row and column permutation.
[ (4th elementary symmetric function of S(n))/(3rd elementary symmetric function of S(n)) ], where S(n) = {first n+3 odd positive integers}.
First differences are 1, 2, 2, 3, 3, 4, 4, 5, 5, ... .
Let M_n denotes the n X n matrix m(i,j) = 1 if i =j; m(i,j) = 1 if (i+j) is odd; m(i,j) = 0 if i+j is even, then a(n) = -det M_(n+1) - Benoit Cloitre, Jun 19 2002
a(n) is the number of squares with corners on an n X n grid, distinct up to translation. See also A002415, A108279.
Starting (1, 3, 5, 8, 11, ...), = row sums of triangle A135841. - Gary W. Adamson, Dec 01 2007
Number of solutions to x+y >= n-1 in integers x,y with 1 <= x <= y <= n-1. - Franz Vrabec, Feb 22 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=5, a(n-4)=-coeff(charpoly(A,x),x^2). - Milan Janjic, Jan 26 2010
Equals row sums of a triangle with alternate columns of (1,2,3,...) and (1,1,1,...). - Gary W. Adamson, May 21 2010
Conjecture: if a(n) = p#(primorial)-1 for some prime number p, then q=(n+1) is also a prime number where p#=floor(q^2/4). Tested up to n=10^100000 no counterexamples are found. It seems that the subsequence is very scattered. So far the triples (p,q,a(q-1)) are {(2,3,1), (3,5,5), (5,11,29), (7,29,209), (17,1429,510509)}. - David Morales Marciel, Oct 02 2015
Numbers of an Ulam spiral starting at 0 in which the shape of the spiral is exactly a rectangle. E.g., a(4)=5 the Ulam spiral is including at that moment only the elements 0,1,2,3,4,5 and the shape is a rectangle. The area is always a(n)+1. E.g., for a(4) the area of the rectangle is 2(rows) X 3(columns) = 6 = a(4) + 1. - David Morales Marciel, Apr 05 2016
Numbers of different quadratic forms (quadrics) in the real projective space P^n(R). - Serkan Sonel, Aug 26 2020
a(n+1) is the number of one-dimensional subspaces of (F_3)^n, counted up to coordinate permutation. E.g.: For n=4, there are five one-dimensional subspaces in (F_3)^3 up to coordinate permutation: [1 2 2] [0 2 2] [1 0 2] [0 0 2] [1 1 1]. This example suggests a bijection (which has to be adjusted for the all-ones matrix) with the binary matrices of the first comment. - Álvar Ibeas, Sep 21 2021
REFERENCES
O. Giering, Vorlesungen über höhere Geometrie, Vieweg, Braunschweig, 1982. See p. 59.
LINKS
Tricia Muldoon Brown, On the unimodality of convolutions of sequences of binomial coefficients, arXiv:1810.08235 [math.CO] (2018). See p. 15.
William F. Lunnon, A postage stamp problem, Comput. J. 12 (1969), 377-380.
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, Vol. 8 (2008), pp. 45-52.
FORMULA
G.f.: x^2*(1+x-x^2)/((1-x^2)*(1-x)^2) = x^2*(1+x-x^2) / ( (1+x)*(1-x)^3 ).
a(n+1) = A002623(n) - A002623(n-1) - 1.
a(n) = A002620(n+1) - 1 = A014616(n-2) + 1.
a(n+1) = A002620(n) + n, n >= 0. - Philippe Deléham, Feb 27 2004
a(0)=0, a(n) = floor(a(n-1) + sqrt(a(n-1)) + 1) for n > 0. - Gerald McGarvey, Jul 30 2004
a(n) = floor((n+1)^2/4) - 1. - Franz Vrabec, Feb 22 2008
a(n) = A005744(n-1) - A005744(n-2). - R. J. Mathar, Nov 04 2008
a(n) = a(n-1) + [side length of the least square > a(n-1) ], that is a(n) = a(n-1) + ceiling(sqrt(a(n-1) + 1)). - Ctibor O. Zizka, Oct 06 2009
For a(1)=0, a(2)=1, a(n) = 2*a(n-1) - a(n-2) + 1 if n is odd; a(n) = 2*a(n-1) - a(n-2) if n is even. - Vincenzo Librandi, Dec 23 2010
a(n) = A181971(n, n-1) for n > 0. - Reinhard Zumkeller, Jul 09 2012
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4); a(1)=0, a(2)=1, a(3)=3, a(4)=5. - Harvey P. Dale, Jun 14 2013
a(n) = floor( (n-1)*(n+3)/4 ). - Wesley Ivan Hurt, Jun 23 2013
a(n) = (2*n^2 + 4*n - 7 - (-1)^n)/8. - Wesley Ivan Hurt, Jul 22 2014
a(n) = a(-n-2) = n-1 + floor( (n-1)^2/4 ). - Bruno Berselli, Feb 03 2015
a(n) = (1/4)*(n+3)^2 - (1/8)*(1 + (-1)^n) - 1. - Serkan Sonel, Aug 26 2020
a(n) + a(n+1) = A034856(n). - R. J. Mathar, Mar 13 2021
a(2*n) = n^2 + n - 1, a(2*n+1) = n^2 + 2*n. - Greg Dresden and Zijie He, Jun 28 2022
Sum_{n>=2} 1/a(n) = 7/4 + tan(sqrt(5)*Pi/2)*Pi/sqrt(5). - Amiram Eldar, Dec 10 2022
E.g.f.: (4 + (x^2 + 3*x - 4)*cosh(x) + (x^2 + 3*x - 3)*sinh(x))/4. - Stefano Spezia, Aug 06 2024
EXAMPLE
There are five 2 X 3 binary matrices with no zero rows or columns up to row and column permutation:
[1 0 0] [1 0 0] [1 1 0] [1 1 0] [1 1 1]
[0 1 1] [1 1 1] [0 1 1] [1 1 1] [1 1 1].
MAPLE
A024206:=n->(2*n^2+4*n-7-(-1)^n)/8: seq(A024206(n), n=1..100);
MATHEMATICA
f[x_, y_] := Floor[ Abs[ y/x - x/y]]; Table[ Floor[ f[2, n^2 + 2 n - 2] /2], {n, 57}] (* Robert G. Wilson v, Aug 11 2010 *)
LinearRecurrence[{2, 0, -2, 1}, {0, 1, 3, 5}, 60] (* Harvey P. Dale, Jun 14 2013 *)
Rest[CoefficientList[Series[x^2 (1 + x - x^2)/((1 - x^2) (1 - x)^2), {x, 0, 70}], x]] (* Vincenzo Librandi, Oct 02 2015 *)
PROG
(PARI) a(n)=(n-1)*(n+3)\4 \\ Charles R Greathouse IV, Jun 26 2013
(PARI) x='x+O('x^99); concat(0, Vec(x^2*(1+x-x^2)/ ((1-x^2)*(1-x)^2))) \\ Altug Alkan, Apr 05 2016
(Haskell)
a024206 n = (n - 1) * (n + 3) `div` 4
a024206_list = scanl (+) 0 $ tail a008619_list
-- Reinhard Zumkeller, Dec 18 2013
(Magma) [(2*n^2+4*n-7-(-1)^n)/8 : n in [1..100]]; // Wesley Ivan Hurt, Jul 22 2014
(GAP) a:=[0, 1, 3, 5];; for n in [5..65] do a[n]:=2*a[n-1]-2*a[n-3]+a[n-4]; od; a; # Muniru A Asiru, Oct 23 2018
(Python) def A024206(n): return (n+1)**2//4 - 1 # Ya-Ping Lu, Jan 01 2024
CROSSREFS
Cf. A014616, A135841, A034856, A005744 (partial sums), A008619 (1st differences).
A row or column of the array A196416 (possibly with 1 subtracted from it).
Cf. A008619.
Second column of A232206.
Sequence in context: A024169 A213706 A078126 * A159325 A228848 A049706
KEYWORD
nonn,easy,nice
EXTENSIONS
Corrected and extended by Vladeta Jovovic, Jun 02 2000
STATUS
approved