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A024206
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Expansion of x^2*(1+x-x^2)/((1-x^2)*(1-x)^2).
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23
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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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}.
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 (benoit7848c(AT)orange.fr), Jun 19 2002
a(n) = A002620(n+1)-1.
a(n) = number of squares with corners on an n X n grid, distinct up to translation. See also A002415, A108279.
Number of solutions to x+y >= n-1 in integers x,y with 1 <= x <= y <= n-1. - Franz Vrabec (franz.vrabec(AT)aon.at), 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). [From Milan R. Janjic (agnus(AT)blic.net), Jan 26 2010]
Equals row sums of a triangle with alternate columns of (1,2,3,...) and (1,1,1,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 21 2010]
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REFERENCES
| W. F. Lunnon, A postage stamp problem. Comput. J. 12 (1969) 377-380. See Table 2.
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LINKS
| Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).
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FORMULA
| First differences are 1, 2, 2, 3, 3, 4, 4, 5, 5, ... .
a(n+1) = A002620(n) + n, n>=0 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 27 2004
a(0)=0, a(n) = floor(a(n-1)+sqrt(a(n-1))+1) for n > 0 - Gerald McGarvey (Gerald.Mcgarvey(AT)comcast.net), Jul 30 2004
Starting (1, 3, 5, 8, 11,...), = row sums of triangle A135841. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 01 2007
a(n) = floor((n+1)^2/4)-1. - Franz Vrabec (franz.vrabec(AT)aon.at), Feb 22 2008
a(n)=A005744(n-1)-A005744(n-2). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 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)). [From Ctibor O. ZIZKA (c.zizka(AT)email.cz), 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.[From Vincenzo Librandi, Dec 23 2010]
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EXAMPLE
| There are 5 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].
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MATHEMATICA
| f[x_, y_] := Floor[ Abs[ y/x - x/y]]; Table[ Floor[ f[2, n^2 + 2 n - 2] /2], {n, 57}] [From Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 11 2010]
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CROSSREFS
| a(n+1)=A002623(n)-A002623(n-1)-1.
Cf. A135841.
Equals A014616 + 1.
A row or column of the array A196416 (possibly with 1 subtracted from it).
Sequence in context: A145197 A024169 A078126 * A159325 A049706 A080415
Adjacent sequences: A024203 A024204 A024205 * A024207 A024208 A024209
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
| Corrected and extended by Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 02 2000
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