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a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n-3*k-1,k) * a(k).
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%I #5 Mar 02 2022 08:47:04

%S 1,1,1,1,1,2,3,4,5,7,10,14,19,26,36,50,69,95,131,181,250,346,482,678,

%T 963,1380,1994,2903,4252,6254,9222,13616,20109,29681,43755,64394,

%U 94583,138632,202755,295906,430986,626585,909500,1318384,1909042,2762122,3994290

%N a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n-3*k-1,k) * a(k).

%F G.f. A(x) satisfies: A(x) = 1 + x * A(x^4/(1 - x)) / (1 - x).

%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 3 k - 1, k] a[k], {k, 0, Floor[(n - 1)/4]}]; Table[a[n], {n, 0, 46}]

%t nmax = 46; A[_] = 0; Do[A[x_] = 1 + x A[x^4/(1 - x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%Y Cf. A119262, A352042.

%K nonn

%O 0,6

%A _Ilya Gutkovskiy_, Mar 01 2022