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 A360168 #18 Apr 10 2024 08:46:53
%S A360168 1,2,6,21,78,297,1145,4447,17358,68001,267141,1051767,4148281,
%T A360168 16385111,64797543,256515731,1016368078,4030114641,15990813773,
%U A360168 63485616391,252175202373,1002136689071,3984080489263,15844839393411,63036297959993,250855287692647
%N A360168 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n,n-3*k).
%F A360168 G.f.: 1 / ( sqrt(1-4*x) * (1 - x^3 * c(x)^6) ), where c(x) is the g.f. of A000108.
%F A360168 D-finite with recurrence n*a(n) +2*(-4*n+3)*a(n-1) +8*(2*n-3)*a(n-2) +3*(-n+2)=0. - _R. J. Mathar_, Mar 12 2023
%F A360168 a(n) = [x^n] 1/(((1-x)^3-x^3) * (1-x)^(n-2)). - _Seiichi Manyama_, Apr 10 2024
%p A360168 A360168 := proc(n)
%p A360168 add(binomial(2*n,n-3*k),k=0..n/3) ;
%p A360168 end proc:
%p A360168 seq(A360168(n),n=0..70) ; # _R. J. Mathar_, Mar 12 2023
%t A360168 a[n_] := Sum[Binomial[2*n, n - 3*k], {k, 0, Floor[n/3]}]; Array[a, 26, 0] (* _Amiram Eldar_, Jan 28 2023 *)
%o A360168 (PARI) a(n) = sum(k=0, n\3, binomial(2*n, n-3*k));
%o A360168 (PARI) my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^3*(2/(1+sqrt(1-4*x)))^6)))
%Y A360168 Cf. A105872, A144904, A360150, A360151, A360152, A360153.
%Y A360168 Cf. A000108, A032443, A114121.
%Y A360168 Cf. A002450, A371777.
%K A360168 nonn
%O A360168 0,2
%A A360168 _Seiichi Manyama_, Jan 28 2023