A261962 Number of compositions of n such that no part equals any of its two immediate predecessors.

1, 1, 1, 3, 3, 5, 11, 15, 23, 37, 67, 101, 165, 265, 419, 691, 1123, 1789, 2909, 4657, 7515, 12183, 19657, 31635, 51101, 82449, 132989, 214623, 346485, 558587, 901399, 1454949, 2347157, 3787197, 6111131, 9858931, 15908393, 25669125, 41416849, 66826277

Offset: 0

Author

Alois P. Heinz, Sep 06 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Column k=2 of A261960.

Cf. A261961.

Programs

Maple

b:= proc(n, i, j) option remember; `if`(n=0, 1, add(
      `if`(k=i or k=j, 0, (t-> b(t, `if`(k>t, 0, k),
      `if`(i>t, 0, i)))(n-k)), k=1..n))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..50);

Mathematica

b[n_, i_, j_] := b[n, i, j] = If[n == 0, 1, Sum[If[k == i || k == j, 0, Function[t, b[t, If[k>t, 0, k], If[i>t, 0, i]]][n - k]], {k, 1, n}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 03 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n, where d = 1.61350953985228953675390530863679475666564394885974..., c = 0.5270561325668460003703909484716134447490733801644227... - Vaclav Kotesovec, Sep 21 2019