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%I #31 Dec 04 2020 15:57:31
%S 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,
%T 259383,0,0,0,12328889,0,0,0,673380980,1659966694,0,0,0,69819104134,0,
%U 0,0,3761284888715,9660240745536,0,0,0,537238185892321,0,0,0,29922345673502904
%N Number of ways the first n squares can be partitioned into three sets with equal sums.
%D Keith F. Lynch, Posting to Math Fun Mailing List, Sep 19 2019.
%H Alois P. Heinz, <a href="/A327450/b327450.txt">Table of n, a(n) for n = 1..57</a>
%F a(n) > 0 => n in { A140282 }. - _Alois P. Heinz_, Sep 29 2019
%e The unique smallest solution (for n = 13) is 1 + 9 + 25 + 36 + 81 + 121 = 16 + 49 + 64 + 144 = 4 + 100 + 169.
%p s:= proc(n) option remember; `if`(n<2, 0, n^2+s(n-1)) end:
%p b:= proc(n, x, y) option remember; `if`(n=1, 1, (p-> (l->
%p add(`if`(p>l[i], 0, b(n-1, sort(subsop(i=l[i]-p, l))
%p [1..2][])), i=1..3))([x, y, s(n)-x-y]))(n^2))
%p end:
%p a:= n-> `if`(irem(1+s(n), 3, 'q')=0, b(n, q-1, q)/2, 0):
%p seq(a(n), n=1..27); # _Alois P. Heinz_, Sep 29 2019
%t s[n_] := s[n] = If[n < 2, 0, n^2 + s[n - 1]];
%t 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}]]];
%t a[n_] := Module[{q, r}, {q, r} = QuotientRemainder[1 + s[n], 3]; If[r == 0, b[n, q - 1, q]/2, 0]];
%t Array[a, 30] (* _Jean-François Alcover_, Dec 04 2020, after _Alois P. Heinz_ *)
%Y Cf. A000290, A000330, A112972, A140282, A275714, A113263, A327448, A327449.
%K nonn
%O 1,17
%A _N. J. A. Sloane_, Sep 20 2019
%E a(28)-a(45) from _Alois P. Heinz_, Sep 29 2019
%E a(46)-a(53) from _Alois P. Heinz_, Oct 05 2019
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