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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.

%I A308682 #13 May 04 2020 12:33:49
%S A308682 1,1,0,0,1,1,1,1,2,7,6,8,13,42,33,52,105,318,310,485,874,3281,2974,
%T A308682 5240,9488,34233,30418,55715,104730,378529,352467,642418,1193879,
%U A308682 4466874,4165910,7762907,14493951,54162165,50621491,95133799,179484713,674845081
%N 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.
%H A308682 Alois P. Heinz, <a href="/A308682/b308682.txt">Table of n, a(n) for n = 0..250</a>
%H A308682 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_problem">Partition problem</a>
%e A308682 a(4) = 1: 1,3,6/10.
%e A308682 a(5) = 1: 1,6,10/3,15.
%e A308682 a(6) = 1: 1,6,21/3,10,15.
%e A308682 a(7) = 1: 1,3,10,28/6,15,21.
%e A308682 a(8) = 2: 1,6,10,15,28/3,21,36; 1,10,21,28/3,6,15,36.
%p A308682 s:= proc(n) s(n):= `if`(n=0, 1, n*(n+1)/2+s(n-1)) end:
%p A308682 b:= proc(n, i) option remember; `if`(i=0, `if`(n<=1, 1, 0),
%p A308682      `if`(n>s(i), 0, (p->b(n+p, i-1)+b(abs(n-p), i-1))(i*(i+1)/2)))
%p A308682     end:
%p A308682 a:= n-> ceil(b(0, n)/2):
%p A308682 seq(a(n), n=0..45);
%t A308682 s[n_] := s[n] = If[n == 0, 1, n(n+1)/2 + s[n-1]];
%t A308682 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]]];
%t A308682 a[n_] := Ceiling[b[0, n]/2];
%t A308682 a /@ Range[0, 45] (* _Jean-François Alcover_, May 04 2020, translated from Maple *)
%Y A308682 Cf. A000217, A000292, A050407, A307877.
%K A308682 nonn
%O A308682 0,9
%A A308682 _Alois P. Heinz_, Jun 16 2019