A240737 Number of compositions of n having exactly two fixed points.

1, 1, 3, 4, 12, 23, 47, 100, 198, 404, 818, 1652, 3319, 6686, 13426, 26947, 54043, 108331, 217059, 434731, 870472, 1742558, 3487710, 6979593, 13965902, 27942597, 55902624, 111833288, 223711791, 447496476, 895110536, 1790410758, 3581127635, 7162749398

Offset: 3

Author

Joerg Arndt and Alois P. Heinz, Apr 11 2014

Keywords

• nonn

Examples

			a(5) = 3: 113, 122, 1211.
a(6) = 4: 1131, 1212, 1221, 12111.
a(7) = 12: 124, 133, 223, 1114, 1132, 1213, 1222, 11311, 12112, 12121, 12211, 121111.
		

Links

• Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 3..1000
• M. Archibald, A. Blecher, and A. Knopfmacher, Fixed Points in Compositions and Words, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.

Crossrefs

Column k=2 of A238349 and of A238350.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, series(
      add(b(n-j, i+1)*`if`(i=j, x, 1), j=1..n), x, 3))
    end:
a:= n-> coeff(b(n, 1), x, 2):
seq(a(n), n=3..40);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, Series[Sum[b[n-j, i+1]*If[i == j, x, 1], {j, 1, n}], {x, 0, 3}]]; a[n_] := SeriesCoefficient[b[n, 1], {x, 0, 2}]; Table[a[n], {n, 3, 40}] (* Jean-François Alcover, Nov 07 2014, after Maple *)

Formula

a(n) ~ c * 2^n, where c = 0.10426192955737153473390619611670767950197436882607451088699497466613223911... . - Vaclav Kotesovec, Sep 07 2014