A261960 Number A(n,k) of compositions of n such that no part equals any of its k immediate predecessors; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 3, 8, 1, 1, 1, 3, 4, 16, 1, 1, 1, 3, 3, 7, 32, 1, 1, 1, 3, 3, 5, 14, 64, 1, 1, 1, 3, 3, 5, 11, 23, 128, 1, 1, 1, 3, 3, 5, 11, 15, 39, 256, 1, 1, 1, 3, 3, 5, 11, 13, 23, 71, 512, 1, 1, 1, 3, 3, 5, 11, 13, 19, 37, 124, 1024

Offset: 0

Author

Alois P. Heinz, Sep 06 2015

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
:  1,  1,  1,  1,  1,  1,  1, ...
:  1,  1,  1,  1,  1,  1,  1, ...
:  2,  1,  1,  1,  1,  1,  1, ...
:  4,  3,  3,  3,  3,  3,  3, ...
:  8,  4,  3,  3,  3,  3,  3, ...
: 16,  7,  5,  5,  5,  5,  5, ...
: 32, 14, 11, 11, 11, 11, 11, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..50, flattened

Crossrefs

Columns k=0-2 give: A011782, A003242, A261962.

Main diagonal gives A032020.

Cf. A261959, A261981.

Programs

Maple

b:= proc(n, l) option remember;
      `if`(n=0, 1, add(`if`(j in l, 0, b(n-j,
      `if`(l=[], [], [subsop(1=NULL, l)[], j]))), j=1..n))
    end:
A:= (n, k)-> b(n, [0$min(n, k)]):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

b[n_, l_] := b[n, l] = If[n==0, 1, Sum[If[MemberQ[l, j], 0, b[n-j, If[l == {}, {}, Append[Rest[l], j]]]], {j, 1, n}]]; A[n_, k_] := b[n, Array[0&, Min[n, k]]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 08 2017, translated from Maple *)