A105872 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-3*k, n).
1, 2, 6, 21, 75, 273, 1009, 3770, 14202, 53846, 205216, 785460, 3017106, 11624580, 44905518, 173863965, 674506059, 2621371005, 10203609597, 39773263035, 155231706951, 606554343495, 2372544034143, 9289131196485, 36401388236461
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
Table[Sum[Binomial[2n-3k,n],{k,0,Floor[n/2]}],{n,0,30}] (* Harvey P. Dale, Jan 13 2015 *) -
PARI
a(n) = sum(k=0, n\3, binomial(2*n-3*k, n)); \\ Seiichi Manyama, Jan 28 2023
Formula
G.f.: 2/(4*x^2+sqrt(1-4*x)*(3*x+1)-5*x+1). - Vladimir Kruchinin, May 24 2014
Conjecture: -3*(n+1)*(7*n-2)*a(n) +6*(7*n+5)*(2*n-1)*a(n-1) -(n+1)*(7*n-2)*a(n-2) +2*(7*n+5)*(2*n-1)*a(n-3)=0. - R. J. Mathar, Nov 28 2014
a(n) ~ 2^(2*n+3) / (7*sqrt(Pi*n)). - Vaclav Kotesovec, Jan 28 2023
a(n) = [x^n] 1/((1-x^3) * (1-x)^(n+1)). - Seiichi Manyama, Apr 08 2024
Extensions
Erroneous title changed by Paul Barry, Apr 14 2010
Name corrected by Seiichi Manyama, Jan 28 2023