A295946 Number of set partitions of [3n] into n subsets of size three such that all element sums are triangular numbers.

1, 1, 1, 2, 17, 149, 1439, 16993, 373393, 6899469, 214746442, 7287992421

Offset

0,4

Links

Table of n, a(n) for n=0..11.

Example

a(0) = 1: {}.
a(1) = 1: {{1,2,3}}.
a(2) = 1: {{1,2,3}, {4,5,6}}.
a(3) = 2: {{1,6,8}, {2,4,9}, {3,5,7}}, {{1,5,9}, {2,6,7}, {3,4,8}}.
a(4) = 17: {{1,2,12}, {3,8,10}, {4,6,11}, {5,7,9}}, {{1,2,12}, {3,7,11}, {4,8,9}, {5,6,10}}, {{1,3,11}, {2,9,10}, {4,5,12}, {6,7,8}}, {{1,6,8}, {2,9,10}, {3,7,11}, {4,5,12}}, {{1,9,11}, {2,3,10}, {4,5,12}, {6,7,8}}, {{1,9,11}, {2,6,7}, {3,8,10}, {4,5,12}}, {{1,4,10}, {2,8,11}, {3,6,12}, {5,7,9}}, {{1,5,9}, {2,8,11}, {3,6,12}, {4,7,10}}, {{1,9,11}, {2,5,8}, {3,6,12}, {4,7,10}}, {{1,3,11}, {2,7,12}, {4,8,9}, {5,6,10}}, {{1,5,9}, {2,7,12}, {3,8,10}, {4,6,11}}, {{1,9,11}, {2,7,12}, {3,4,8}, {5,6,10}}, {{1,9,11}, {2,7,12}, {3,8,10}, {4,5,6}}, {{1,8,12}, {2,3,10}, {4,6,11}, {5,7,9}}, {{1,8,12}, {2,9,10}, {3,5,7}, {4,6,11}}, {{1,8,12}, {2,4,9}, {3,7,11}, {5,6,10}}, {{1,8,12}, {2,9,10}, {3,7,11}, {4,5,6}}.

Maple

b:= proc(s) option remember; `if`(s={}, 1, (j->
      add(add(`if`(i<j and k<i and issqr((k+i+j)*8+1),
       b(s minus {k, i, j}), 0), k=s), i=s))(max(s)))
    end:
a:= n-> b({$1..3*n}):
seq(a(n), n=0..7);

Mathematica

b[s_] := b[s] = If[s == {}, 1, With[{j = Max[s]},
     Sum[Sum[If[i < j && k < i && IntegerQ@Sqrt[(k + i + j)*8 + 1],
     b[s ~Complement~ {k, i, j}], 0], {k, s}], {i, s}]]];
a[n_] := b[Range[3n]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 9}] (* Jean-François Alcover, Mar 08 2021, after Alois P. Heinz *)

Crossrefs

Cf. A000217, A278329.

Sequence in context: A273340 A074624 A245807 * A176934 A126037 A241135

Adjacent sequences: A295943 A295944 A295945 * A295947 A295948 A295949

Keyword

nonn,more

Author

Alois P. Heinz, Nov 30 2017

Status

approved