A319398 -id:A319398 - 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).

A357690 Number of ways to write n as an ordered sum of five positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 5, 15, 30, 50, 71, 95, 115, 140, 165, 191, 205, 220, 240, 260, 285, 310, 325, 325, 320, 341, 350, 380, 385, 405, 420, 430, 450, 465, 465, 445, 410, 435, 425, 450, 481, 495, 515, 490, 510, 555, 525, 580, 540, 530, 570, 530, 580, 600, 520, 525, 440, 455, 520, 445, 555, 530

Offset: 5

Ilya Gutkovskiy, Oct 10 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..10000

Cf. A000045, A076739, A121548, A121549, A121550, A319398, A357688, A357691.

nmax = 61; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 2, 21}]^5, {x, 0, nmax}], x] // Drop[#, 5] &

G.f.: ( Sum_{k>=2} x^Fibonacci(k) )^5.

a(n) = A121548(n,5).

A358005 Number of partitions of n into 5 distinct positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

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

Offset: 19

Ilya Gutkovskiy, Oct 24 2022

nonn

Robert Israel, Table of n, a(n) for n = 19..10000

Cf. A000045, A000119, A319398, A357690, A357722, A357731, A357732, A358006, A358007, A358008.

G:= mul(1+t*x^combinat:-fibonacci(k),k=2..21):
S:= coeff(expand(G),t,5):
seq(coeff(S,x,i),i=19..110); # Robert Israel, Jan 06 2023

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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A357690 Number of ways to write n as an ordered sum of five positive Fibonacci numbers (with a single type of 1).

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Mathematica

Formula

A358005 Number of partitions of n into 5 distinct positive Fibonacci numbers (with a single type of 1).

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Maple