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

1,17

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

Formula

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A139510 A142651 A214265 * A142135 A142257 A141926

Adjacent sequences: A327447 A327448 A327449 * A327451 A327452 A327453

Keyword

nonn

Author

N. J. A. Sloane, Sep 20 2019

Extensions

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

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

Status

approved