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A387516 a(n) = Sum_{k=0..floor(n/3)} 2^(n-3*k) * binomial(k,n-3*k)^2.

%I A387516 #16 Sep 01 2025 12:19:45
%S A387516 1,0,0,1,2,0,1,8,4,1,18,36,9,32,144,129,66,400,801,472,932,3201,3698,
%T A387516 2916,9865,19728,17248,28225,78690,105536,106625,262408,516388,566785,
%U A387516 871730,2064964,3040713,3585888,7366032,14098817,17860962,27066384,56844833,88593688
%N A387516 a(n) = Sum_{k=0..floor(n/3)} 2^(n-3*k) * binomial(k,n-3*k)^2.
%H A387516 Vincenzo Librandi, <a href="/A387516/b387516.txt">Table of n, a(n) for n = 0..2000</a>
%F A387516 G.f.: 1/sqrt((1-x^3-2*x^4)^2 - 8*x^7).
%t A387516 Table[Sum[2^(n-3*k)*Binomial[k,n-3*k]^2,{k,0,Floor[n/3]}],{n,0,40}] (* _Vincenzo Librandi_, Sep 01 2025 *)
%o A387516 (PARI) a(n) = sum(k=0, n\3, 2^(n-3*k)*binomial(k, n-3*k)^2);
%o A387516 (Magma) [(&+[2^(n-3*k) * Binomial(k,n-3*k)^2: k in [0..Floor(n/2)]]): n in [0..40]]; // _Vincenzo Librandi_, Sep 01 2025
%Y A387516 Cf. A387484, A387485.
%Y A387516 Cf. A387482.
%K A387516 nonn,new
%O A387516 0,5
%A A387516 _Seiichi Manyama_, Sep 01 2025