This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A360274 #13 Mar 12 2023 11:04:07
%S A360274 1,1,2,6,15,44,138,444,1474,5000,17240,60260,213012,760140,2734700,
%T A360274 9907857,36117810,132379490,487546557,1803381000,6696499910,
%U A360274 24953813577,93285944640,349756113560,1314857960901,4955232346092,18717109185712,70848408876905
%N A360274 a(n) = Sum_{k=0..floor(n/3)} Catalan(n-3*k).
%F A360274 G.f.: c(x)/(1-x^3), where c(x) is the g.f. of A000108.
%F A360274 a(n) ~ 2^(2*n+6) / (63*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Feb 01 2023
%F A360274 D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(-n-1)*a(n-3) +2*(2*n-1)*a(n-4)=0. - _R. J. Mathar_, Mar 12 2023
%p A360274 A360274 := proc(n)
%p A360274 add(A000108(n-3*k),k=0..n/3) ;
%p A360274 end proc:
%p A360274 seq(A360274(n),n=0..70) ; # _R. J. Mathar_, Mar 12 2023
%o A360274 (PARI) a(n) = sum(k=0, n\3, binomial(2*(n-3*k), n-3*k)/(n-3*k+1));
%o A360274 (PARI) my(N=30, x='x+O('x^N)); Vec(2/((1-x^3)*(1+sqrt(1-4*x))))
%Y A360274 Cf. A014137, A360273.
%Y A360274 Cf. A000108.
%K A360274 nonn,easy
%O A360274 0,3
%A A360274 _Seiichi Manyama_, Jan 31 2023