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A240736
Number of compositions of n having exactly one fixed point.
9
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
KEYWORD
nonn
AUTHOR
Joerg Arndt and Alois P. Heinz, Apr 11 2014
STATUS
approved