A307877 Number of ways of partitioning the set of the first n positive squares into two subsets whose sums differ at most by 1.

1, 1, 0, 0, 0, 0, 2, 1, 1, 5, 2, 1, 5, 13, 43, 43, 57, 193, 274, 239, 430, 1552, 3245, 2904, 5419, 18628, 31048, 27813, 50213, 188536, 372710, 348082, 649300, 2376996, 4197425, 3913496, 7287183, 27465147, 53072709, 50030553, 93696497, 351329160, 650125358

Offset: 0

Author

Alois P. Heinz, Jun 04 2019

Keywords

• nonn

Examples

			a(6) = 2: 1,9,36/4,16,25; 1,4,16,25/9,36.
a(7) = 1: 1,4,16,49/9,25,36.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• Wikipedia, Partition problem

Crossrefs

Cf. A000290, A069918, A083527, A158092, A306443, A308682.

Programs

Maple

s:= proc(n) s(n):= `if`(n=0, 1, n^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^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^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^2]]];
a[n_] := Ceiling[b[0, n]/2];
a /@ Range[0, 45] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

a(n) = A083527(n) if n == 0 or 3 (mod 4).