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A359513
Number of partitions of n into at most 4 positive Fibonacci numbers (with a single type of 1).
5
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, 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, 6, 3, 10, 5, 7, 9, 5, 8, 5, 2, 9, 4, 7, 6, 2, 8, 4, 3, 8, 1, 4, 1
OFFSET
0,3
LINKS
FORMULA
a(n) = Sum_{k=0..4} A319394(n,k). - Alois P. Heinz, Jan 03 2023
MAPLE
h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
end:
a:= n-> (p-> add(coeff(p, x, i), i=0..4))(b(n, h(n))):
seq(a(n), n=0..87); # Alois P. Heinz, Jan 03 2023
MATHEMATICA
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]]]];
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]]]]];
a[n_] := Sum[Coefficient[#, x, i], {i, 0, 4}]&[b[n, h[n]]];
Table[a[n], {n, 0, 87}] (* Jean-François Alcover, May 26 2023, after Alois P. Heinz *)
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jan 03 2023
STATUS
approved