A344707 Number of partitions of the n-th triangular number n*(n+1)/2 into a triangular number of triangular parts.

1, 1, 2, 2, 4, 6, 10, 17, 27, 59, 116, 224, 427, 839, 1616, 3110, 5968, 11337, 21560, 41002, 77058, 144395, 270041, 500683, 926185, 1706838, 3126023, 5706968, 10379752, 18782112, 33868690, 60857056, 108867121, 194086938, 344821828, 610237587, 1076401704

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..141

Example

a(5) = 6: [15], [10,1,1,1,1,1], [6,6,3], [6,3,3,1,1,1], [6,1,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1].

Maple

t:= n-> issqr(8*n+1):
h:= proc(n) option remember; `if`(t(n), n, h(n-1)) end:
b:= proc(n, i, c) option remember; `if`(n=0 or i=1, `if`(
      t(c+n), 1, 0), b(n-i, h(min(n-i, i)), c+1)+b(n, h(i-1), c))
    end:
a:= n-> b(n*(n+1)/2$2, 0):
seq(a(n), n=0..40);

Mathematica

t[n_] := IntegerQ@Sqrt[8n+1];
h[n_] := h[n] = If[t[n], n, h[n-1]];
b[n_, i_, c_] := b[n, i, c] = If[n == 0 || i == 1, If[t[c+n], 1, 0], b[n-i, h[Min[n-i, i]], c+1] + b[n, h[i-1], c]];
a[n_] := b[n(n+1)/2, n(n+1)/2, 0];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jan 10 2023, after Alois P. Heinz *)

Crossrefs

Cf. A000217.

Sequence in context: A361223 A060553 A293673 * A293505 A032307 A007560

Adjacent sequences: A344704 A344705 A344706 * A344708 A344709 A344710

Keyword

nonn

Author

Alois P. Heinz, May 26 2021

Status

approved