A359512 -id:A359512 - cp's OEIS frontend

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

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

Ilya Gutkovskiy, Jan 03 2023

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A000045, A003107, A319394, A319397, A359511, A359512, A359516.

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

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 *)

a(n) = Sum_{k=0..4} A319394(n,k). - Alois P. Heinz, Jan 03 2023

A359511 Number of partitions of n into at most 2 positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 0, 2, 1, 1, 2, 0, 1, 0, 0, 2, 1, 1, 1, 0, 2, 0, 0, 1, 0, 0, 0, 0, 2, 1, 1, 1, 0, 1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1, 0, 1, 0, 0, 1

Offset: 0

Ilya Gutkovskiy, Jan 03 2023

nonn

Cf. A000045, A003107, A319394, A319395, A359512, A359513, A359514.

a(n) = Sum_{k=0..2} A319394(n,k). - Alois P. Heinz, Jan 03 2023

A359515 Number of compositions (ordered partitions) of n into at most 3 positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 10, 11, 12, 12, 12, 14, 12, 12, 11, 12, 15, 12, 14, 12, 6, 12, 8, 14, 15, 9, 15, 12, 9, 14, 6, 12, 6, 0, 12, 8, 11, 17, 9, 15, 9, 6, 15, 9, 12, 9, 0, 14, 6, 6, 12, 0, 6, 0, 0, 12, 8, 11, 14, 9, 17, 9, 6, 15, 6, 9, 6, 0, 15, 9, 9, 12, 0, 9, 0, 0, 14, 6, 6, 6, 0, 12

Offset: 0

Ilya Gutkovskiy, Jan 03 2023

nonn

Cf. A000045, A076739, A121548, A121550, A359512, A359514, A359516.

g:= proc(n) g(n):= (t-> issqr(t+4) or issqr(t-4))(5*n^2) end:
b:= proc(n, t) option remember; `if`(n=0, 1, `if`(t<1, 0,
      add(`if`(g(j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 3):
seq(a(n), n=0..81);  # Alois P. Heinz, Jan 03 2023

g[n_] := With[{t = 5 n^2}, IntegerQ @ Sqrt[t+4] || IntegerQ @ Sqrt[t-4]];
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}]]];
a[n_] :=  b[n, 3];
Table[a[n], {n, 0, 81}] (* Jean-François Alcover, May 28 2023, after Alois P. Heinz *)

a(n) = Sum_{k=0..3} A121548(n,k). - Alois P. Heinz, Jan 03 2023

A359538 Number of partitions of n into at most 3 distinct positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 1, 3, 2, 2, 2, 1, 3, 2, 2, 3, 1, 2, 1, 1, 3, 2, 2, 3, 1, 3, 1, 1, 2, 1, 1, 1, 0, 3, 2, 2, 3, 1, 3, 1, 1, 3, 1, 1, 1, 0, 2, 1, 1, 1, 0, 1, 0, 0, 3, 2, 2, 3, 1, 3, 1, 1, 3, 1, 1, 1, 0, 3, 1, 1, 1, 0, 1, 0, 0, 2, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 3, 2, 2, 3, 1, 3

Offset: 0

Ilya Gutkovskiy, Jan 04 2023

nonn

Cf. A000045, A000119, A357732, A359512, A359515, A359537, A359539.

cp's OEIS Frontend

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

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A359511 Number of partitions of n into at most 2 positive Fibonacci numbers (with a single type of 1).

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A359515 Number of compositions (ordered partitions) of n into at most 3 positive Fibonacci numbers (with a single type of 1).

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A359538 Number of partitions of n into at most 3 distinct positive Fibonacci numbers (with a single type of 1).

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