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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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
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
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Jun 07 2012
STATUS
approved

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Last modified June 19 03:10 EDT 2024. Contains 373492 sequences. (Running on oeis4.)