A213238 Triangle T(n,k) in which n-th row lists in increasing order the distinct values v satisfying v = sum of elements in S = product of elements in P for a partition of {1,...,n} into two sets S and P.

1, 3, 8, 12, 18, 24, 32, 40, 42, 50, 60, 64, 72, 84, 88, 90, 98, 99, 105, 112, 120, 128, 130, 135, 144, 162, 168, 180, 182, 192, 200, 208, 210, 220, 231, 242, 252, 264, 266, 272, 280, 288, 294, 300, 312, 315, 320, 324, 330, 338, 340, 360, 364, 378, 392, 400

Offset

1,2

Links

Alois P. Heinz, Rows n = 1..798, flattened

Formula

T(n,1) = floor((n-1)^2/2) = A007590(n-1) for n>=5.

Example

For n=1 v=1 satisfies the condition with S={1}, P={} => row 1 = [1].
For n=2 no v can be found => row 2 is empty: [].
For n=3 there is one solution: S={1,2}, P={3}, v=3 => row 3 = [3].
For n=10 we have three partitions of {1,2,...,10} into S and P satisfying v = Sum_{i:S} i = Product_{k:P} k but there are only two distinct values v: S={2,3,5,6,7,8,9}, P={1,4,10}, v=40; S={4,5,6,8,9,10}, P={1,2,3,7}, v=42; S={1,2,3,4,5,8,9,10}, P={6,7}, v=42 => row 10 = [40, 42].
Triangle T begins:
   1;
    ;
   3;
    ;
   8;
  12;
  18;
  24;
  32;
  40, 42;
  50;
  60, 64;
  72;
  84, 88, 90;
  ...

Maple

b:= proc(n, s, p)
      `if`(s=p, {s}, `if`(n<1, {}, {b(n-1, s, p)[],
      `if`(s-n<p*n, {}, b(n-1, s-n, p*n))[]}))
    end:
T:= n-> sort([b(n, n*(n+1)/2, 1)[]])[]:
seq(T(n), n=1..30);

Mathematica

b[n_, s_, p_] :=
     If[s == p, {s}, If[n < 1, {}, {b[n-1, s, p],
     If[s-n < p*n, {}, b[n-1, s-n, p*n]]} // Union]];
T[n_] := Sort[b[n, n(n+1)/2, 1] // Flatten] // Union;
Array[T, 30] // Flatten (* Jean-François Alcover, Feb 19 2021, after Alois P. Heinz *)

Crossrefs

Row lengths (or number of solutions) are in A213237.

T(n,1) = A007590(n-1) for n>=5.

Sequence in context: A256711 A219635 A063225 * A005660 A086813 A287081

Adjacent sequences: A213235 A213236 A213237 * A213239 A213240 A213241

Keyword

nonn,tabf

Author

Alois P. Heinz, Jun 07 2012

Status

approved