A327450 Number of ways the first n squares can be partitioned into three sets with equal sums.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 137, 211, 0, 0, 0, 3035, 0, 0, 0, 120465, 259383, 0, 0, 0, 12328889, 0, 0, 0, 673380980, 1659966694, 0, 0, 0, 69819104134, 0, 0, 0, 3761284888715, 9660240745536, 0, 0, 0, 537238185892321, 0, 0, 0, 29922345673502904

Offset: 1

Author

N. J. A. Sloane, Sep 20 2019

Keywords

• nonn

Examples

			The unique smallest solution (for n = 13) is 1 + 9 + 25 + 36 + 81 + 121 = 16 + 49 + 64 + 144 = 4 + 100 + 169.
		

References

• Keith F. Lynch, Posting to Math Fun Mailing List, Sep 19 2019.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..57

Crossrefs

Cf. A000290, A000330, A112972, A140282, A275714, A113263, A327448, A327449.

Programs

Maple

s:= proc(n) option remember; `if`(n<2, 0, n^2+s(n-1)) end:
b:= proc(n, x, y) option remember; `if`(n=1, 1, (p-> (l->
      add(`if`(p>l[i], 0, b(n-1, sort(subsop(i=l[i]-p, l))
             [1..2][])), i=1..3))([x, y, s(n)-x-y]))(n^2))
    end:
a:= n-> `if`(irem(1+s(n), 3, 'q')=0, b(n, q-1, q)/2, 0):
seq(a(n), n=1..27);  # Alois P. Heinz, Sep 29 2019

Mathematica

s[n_] := s[n] = If[n < 2, 0, n^2 + s[n - 1]];
b[n_, x_, y_] := b[n, x, y] = Module[{p, l}, If[n == 1, 1, p = n^2; l = {x, y, s[n] - x - y}; Sum[If[p > l[[i]], 0, b[n - 1, Sequence @@ Sort[ ReplacePart[l, i -> l[[i]] - p]][[1 ;; 2]]]], {i, 1, 3}]]];
a[n_] := Module[{q, r}, {q, r} = QuotientRemainder[1 + s[n], 3]; If[r == 0, b[n, q - 1, q]/2, 0]];
Array[a, 30] (* Jean-François Alcover, Dec 04 2020, after Alois P. Heinz *)

Formula

a(n) > 0 => n in { A140282 }. - Alois P. Heinz, Sep 29 2019

Extensions

a(28)-a(45) from Alois P. Heinz, Sep 29 2019

a(46)-a(53) from Alois P. Heinz, Oct 05 2019