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

1,4

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.

Formula

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

Example

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

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 *)

Crossrefs

Column k=1 of A238349 and of A238350.

Sequence in context: A025619 A093210 A133600 * A286741 A298344 A285654

Adjacent sequences: A240733 A240734 A240735 * A240737 A240738 A240739

Keyword

nonn

Author

Joerg Arndt and Alois P. Heinz, Apr 11 2014

Status

approved