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A359514 Number of compositions (ordered partitions) of n into at most 2 positive Fibonacci numbers (with a single type of 1).

%I A359514 #12 May 26 2023 03:50:21
%S A359514 1,1,2,3,3,3,3,2,3,2,3,2,0,3,2,2,3,0,2,0,0,3,2,2,2,0,3,0,0,2,0,0,0,0,
%T A359514 3,2,2,2,0,2,0,0,3,0,0,0,0,2,0,0,0,0,0,0,0,3,2,2,2,0,2,0,0,2,0,0,0,0,
%U A359514 3,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,2,2,2,0,2
%N A359514 Number of compositions (ordered partitions) of n into at most 2 positive Fibonacci numbers (with a single type of 1).
%H A359514 Alois P. Heinz, <a href="/A359514/b359514.txt">Table of n, a(n) for n = 0..20000</a>
%F A359514 a(n) = Sum_{k=0..2} A121548(n,k). - _Alois P. Heinz_, Jan 03 2023
%p A359514 g:= proc(n) g(n):= (t-> issqr(t+4) or issqr(t-4))(5*n^2) end:
%p A359514 b:= proc(n, t) option remember; `if`(n=0, 1, `if`(t<1, 0,
%p A359514       add(`if`(g(j), b(n-j, t-1), 0), j=1..n)))
%p A359514     end:
%p A359514 a:= n-> b(n, 2):
%p A359514 seq(a(n), n=0..94);  # _Alois P. Heinz_, Jan 03 2023
%t A359514 g[n_] := IntegerQ@Sqrt[# + 4] || IntegerQ@Sqrt[# - 4]&[5 n^2];
%t A359514 b[n_, t_] := b[n, t] = If[n == 0, 1, If[t < 1, 0, Sum[If[g[j], b[n - j, t - 1], 0], {j, 1, n}]]];
%t A359514 a[n_] := b[n, 2];
%t A359514 Table[a[n], {n, 0, 94}] (* _Jean-François Alcover_, May 26 2023, after _Alois P. Heinz_ *)
%Y A359514 Cf. A000045, A076739, A121548, A121549, A359511, A359515, A359516.
%K A359514 nonn
%O A359514 0,3
%A A359514 _Ilya Gutkovskiy_, Jan 03 2023