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

0,9

Links

Alois P. Heinz, Table of n, a(n) for n = 0..250

Wikipedia, Partition problem

Example

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.

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 *)

Crossrefs

Cf. A000217, A000292, A050407, A307877.

Sequence in context: A156591 A233770 A138283 * A117968 A217568 A320871

Adjacent sequences: A308679 A308680 A308681 * A308683 A308684 A308685

Keyword

nonn

Author

Alois P. Heinz, Jun 16 2019

Status

approved