A024206 Expansion of x^2*(1+x-x^2)/((1-x^2)*(1-x)^2).
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
Examples
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].
References
- O. Giering, Vorlesungen über höhere Geometrie, Vieweg, Braunschweig, 1982. See p. 59.
Links
- Muniru A Asiru, Table of n, a(n) for n = 1..5000
- Tricia Muldoon Brown, On the unimodality of convolutions of sequences of binomial coefficients, arXiv:1810.08235 [math.CO] (2018). See p. 15.
- Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
- 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.
- Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Crossrefs
Programs
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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
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Haskell
a024206 n = (n - 1) * (n + 3) `div` 4 a024206_list = scanl (+) 0 $ tail a008619_list -- Reinhard Zumkeller, Dec 18 2013
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Magma
[(2*n^2+4*n-7-(-1)^n)/8 : n in [1..100]]; // Wesley Ivan Hurt, Jul 22 2014
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Maple
A024206:=n->(2*n^2+4*n-7-(-1)^n)/8: seq(A024206(n), n=1..100);
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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}] (* 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 *) -
PARI
a(n)=(n-1)*(n+3)\4 \\ Charles R Greathouse IV, Jun 26 2013
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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 -
Python
def A024206(n): return (n+1)**2//4 - 1 # Ya-Ping Lu, Jan 01 2024
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) = 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) = 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
Extensions
Corrected and extended by Vladeta Jovovic, Jun 02 2000
Comments