A240736 Number of compositions of n having exactly one fixed point.

1, 1, 1, 4, 7, 16, 29, 60, 120, 238, 479, 956, 1910, 3817, 7633, 15252, 30491, 60955, 121865, 243650, 487165, 974112, 1947851, 3895086, 7789153, 15576624, 31150481, 62296424, 124585395, 249158607, 498297297, 996562085, 1993071152, 3986055928, 7971971230

Offset: 1

Author

Joerg Arndt and Alois P. Heinz, Apr 11 2014

Keywords

• nonn

Examples

			a(4) = 4: 13, 22, 112, 1111.
a(5) = 7: 14, 32, 131, 221, 1112, 1121, 11111.
		

References

• M. Archibald, A. Blecher and A. Knopfmacher, Fixed points in compositions and words, accepted by the Journal of Integer Sequences.

Links

• Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 1..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=1 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, 2))
    end:
a:= n-> coeff(b(n, 1), x, 1):
seq(a(n), n=1..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, 2}]]; a[n_] := SeriesCoefficient[b[n, 1], {x, 0, 1}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Nov 06 2014, after Maple *)

Formula

a(n) ~ c * 2^n, where c = A065442 * A048651 / 2 = 0.2319972162254452238942023675457837005318389885... - Vaclav Kotesovec, Sep 06 2014