A319402 -id:A319402 - cp's OEIS frontend

A319394 Number T(n,k) of partitions of n into exactly k positive Fibonacci numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 2, 1, 1, 0, 1, 1, 2, 1, 1, 0, 0, 2, 2, 2, 1, 1, 0, 0, 1, 3, 2, 2, 1, 1, 0, 1, 1, 2, 4, 2, 2, 1, 1, 0, 0, 1, 3, 3, 4, 2, 2, 1, 1, 0, 0, 2, 2, 4, 4, 4, 2, 2, 1, 1, 0, 0, 1, 3, 4, 5, 4, 4, 2, 2, 1, 1, 0, 0, 0, 3, 5, 5, 6, 4, 4, 2, 2, 1, 1

Offset: 0

Alois P. Heinz, Sep 18 2018

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k <= n. T(n,k) = 0 for k > n.

			T(14,3) = 2: 851, 833.
T(14,4) = 5: 8321, 8222, 5531, 5522, 5333.
T(14,5) = 6: 83111, 82211, 55211, 53321, 53222, 33332.
T(14,6) = 8: 821111, 551111, 533111, 532211, 522221, 333311, 333221, 332222.
T(14,7) = 7: 8111111, 5321111, 5222111, 3332111, 3322211, 3222221, 2222222.
T(14,8) = 6: 53111111, 52211111, 33311111, 33221111, 32222111, 22222211.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1, 1;
  0, 0, 2, 1, 1;
  0, 1, 1, 2, 1, 1;
  0, 0, 2, 2, 2, 1, 1;
  0, 0, 1, 3, 2, 2, 1, 1;
  0, 1, 1, 2, 4, 2, 2, 1, 1;
  0, 0, 1, 3, 3, 4, 2, 2, 1, 1;
  0, 0, 2, 2, 4, 4, 4, 2, 2, 1, 1;
  0, 0, 1, 3, 4, 5, 4, 4, 2, 2, 1, 1;
  0, 0, 0, 3, 5, 5, 6, 4, 4, 2, 2, 1, 1;
  0, 1, 1, 2, 4, 7, 6, 6, 4, 4, 2, 2, 1, 1;
  0, 0, 1, 2, 5, 6, 8, 7, 6, 4, 4, 2, 2, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give: A000007, A010056 (for n>0), A319395, A319396, A319397, A319398, A319399, A319400, A319401, A319402, A319403.
Row sums give A003107.
T(2n,n) gives A136343.
Cf. A000045, A281689, A319503.

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:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, h(n))):
seq(T(n), n=0..20);

T[n_, k_] := SeriesCoefficient[1/Product[(1 - y x^Fibonacci[j]) + O[x]^(n+1) // Normal, {j, 2, n+1}], {x, 0, n}, {y, 0, k}];
Table[T[n, k], {n, 0, 40}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 28 2020 *)
h[n_] := h[n] = If[n < 1, 0, If[Function[t, IntegerQ@Sqrt[t + 4] || IntegerQ@Sqrt[t - 4]][5 n^2], n, h[n - 1]]];
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1 || t < 1, 0, b[n, h[i - 1], t] + b[n - i, h[Min[n - i, i]], t - 1]]];
T[n_, k_] :=  b[n, h[n], k] - b[n, h[n], k - 1];
Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

T(n,k) = [x^n y^k] 1/Product_{j>=2} (1-y*x^A000045(j)).
Sum_{k=1..n} k * T(n,k) = A281689(n).
T(A000045(n),n) = A319503(n).

A357717 Number of ways to write n as an ordered sum of nine positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 9, 45, 156, 423, 954, 1878, 3321, 5409, 8251, 11979, 16686, 22446, 29250, 37134, 46107, 56259, 67671, 80407, 94338, 109269, 125118, 141930, 159723, 178608, 198522, 219510, 241338, 264438, 288810, 314550, 341010, 367785, 394596, 421443, 448650, 476614, 505404, 534978

Offset: 9

Ilya Gutkovskiy, Oct 10 2022

nonn

Cf. A000045, A076739, A121548, A121549, A121550, A319402, A357688, A357690, A357691, A357694, A357716.

nmax = 47; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 2, 21}]^9, {x, 0, nmax}], x] // Drop[#, 9] &

G.f.: ( Sum_{k>=2} x^Fibonacci(k) )^9.

a(n) = A121548(n,9).

cp's OEIS Frontend

A319394 Number T(n,k) of partitions of n into exactly k positive Fibonacci numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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