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A359513 Number of partitions of n into at most 4 positive Fibonacci numbers (with a single type of 1).

%I A359513 #13 May 26 2023 05:19:14
%S A359513 1,1,2,3,4,5,6,6,8,7,8,8,8,8,8,8,9,8,9,8,8,8,7,8,9,7,10,8,8,9,7,8,8,4,
%T A359513 8,5,8,9,6,10,8,6,10,6,9,8,5,9,6,6,8,4,8,4,1,8,4,7,9,5,10,7,6,10,6,8,
%U A359513 6,3,10,5,7,9,5,8,5,2,9,4,7,6,2,8,4,3,8,1,4,1
%N A359513 Number of partitions of n into at most 4 positive Fibonacci numbers (with a single type of 1).
%H A359513 Alois P. Heinz, <a href="/A359513/b359513.txt">Table of n, a(n) for n = 0..20000</a>
%F A359513 a(n) = Sum_{k=0..4} A319394(n,k). - _Alois P. Heinz_, Jan 03 2023
%p A359513 h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
%p A359513       issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
%p A359513     end:
%p A359513 b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
%p A359513       b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
%p A359513     end:
%p A359513 a:= n-> (p-> add(coeff(p, x, i), i=0..4))(b(n, h(n))):
%p A359513 seq(a(n), n=0..87);  # _Alois P. Heinz_, Jan 03 2023
%t A359513 h[n_] := h[n] = If[n < 1, 0, With[{t = 5 n^2}, If[IntegerQ @ Sqrt[t + 4] || IntegerQ @ Sqrt[t - 4], n, h[n - 1]]]];
%t A359513 b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x^n, b[n, h[i - 1]] + Expand[x*b[n - i, h[Min[n - i, i]]]]];
%t A359513 a[n_] := Sum[Coefficient[#, x, i], {i, 0, 4}]&[b[n, h[n]]];
%t A359513 Table[a[n], {n, 0, 87}] (* _Jean-François Alcover_, May 26 2023, after _Alois P. Heinz_ *)
%Y A359513 Cf. A000045, A003107, A319394, A319397, A359511, A359512, A359516.
%K A359513 nonn
%O A359513 0,3
%A A359513 _Ilya Gutkovskiy_, Jan 03 2023