Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #29 Apr 08 2024 09:12:57
%S 1,2,6,21,75,273,1009,3770,14202,53846,205216,785460,3017106,11624580,
%T 44905518,173863965,674506059,2621371005,10203609597,39773263035,
%U 155231706951,606554343495,2372544034143,9289131196485,36401388236461
%N a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-3*k, n).
%H Seiichi Manyama, <a href="/A105872/b105872.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: 2/(4*x^2+sqrt(1-4*x)*(3*x+1)-5*x+1). - _Vladimir Kruchinin_, May 24 2014
%F 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
%F a(n) ~ 2^(2*n+3) / (7*sqrt(Pi*n)). - _Vaclav Kotesovec_, Jan 28 2023
%F a(n) = [x^n] 1/((1-x^3) * (1-x)^(n+1)). - _Seiichi Manyama_, Apr 08 2024
%t Table[Sum[Binomial[2n-3k,n],{k,0,Floor[n/2]}],{n,0,30}] (* _Harvey P. Dale_, Jan 13 2015 *)
%o (PARI) a(n) = sum(k=0, n\3, binomial(2*n-3*k, n)); \\ _Seiichi Manyama_, Jan 28 2023
%Y Cf. A144904, A360150, A360151, A360152, A360153.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Apr 23 2005
%E Erroneous title changed by _Paul Barry_, Apr 14 2010
%E Name corrected by _Seiichi Manyama_, Jan 28 2023