A308682 Number of ways of partitioning the set of the first n positive triangular numbers into two subsets whose sums differ at most by 1.

1, 1, 0, 0, 1, 1, 1, 1, 2, 7, 6, 8, 13, 42, 33, 52, 105, 318, 310, 485, 874, 3281, 2974, 5240, 9488, 34233, 30418, 55715, 104730, 378529, 352467, 642418, 1193879, 4466874, 4165910, 7762907, 14493951, 54162165, 50621491, 95133799, 179484713, 674845081

Offset: 0

Author

Alois P. Heinz, Jun 16 2019

Keywords

• nonn

Examples

			a(4) = 1: 1,3,6/10.
a(5) = 1: 1,6,10/3,15.
a(6) = 1: 1,6,21/3,10,15.
a(7) = 1: 1,3,10,28/6,15,21.
a(8) = 2: 1,6,10,15,28/3,21,36; 1,10,21,28/3,6,15,36.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250
• Wikipedia, Partition problem

Crossrefs

Cf. A000217, A000292, A050407, A307877.

Programs

Maple

s:= proc(n) s(n):= `if`(n=0, 1, n*(n+1)/2+s(n-1)) end:
b:= proc(n, i) option remember; `if`(i=0, `if`(n<=1, 1, 0),
     `if`(n>s(i), 0, (p->b(n+p, i-1)+b(abs(n-p), i-1))(i*(i+1)/2)))
    end:
a:= n-> ceil(b(0, n)/2):
seq(a(n), n=0..45);

Mathematica

s[n_] := s[n] = If[n == 0, 1, n(n+1)/2 + s[n-1]];
b[n_, i_] := b[n, i] = If[i == 0, If[n <= 1, 1, 0], If[n > s[i], 0, Function[p, b[n + p, i-1] + b[Abs[n-p], i-1]][i(i+1)/2]]];
a[n_] := Ceiling[b[0, n]/2];
a /@ Range[0, 45] (* Jean-François Alcover, May 04 2020, translated from Maple *)