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
Examples
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; ...
Links
- Alois P. Heinz, Rows n = 0..200, flattened
Crossrefs
Programs
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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: T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, h(n))): seq(T(n), n=0..20); -
Mathematica
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 *)
Comments