login
A240737
Number of compositions of n having exactly two fixed points.
3
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
3,3
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.
FORMULA
a(n) ~ c * 2^n, where c = 0.10426192955737153473390619611670767950197436882607451088699497466613223911... . - Vaclav Kotesovec, Sep 07 2014
EXAMPLE
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.
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 *)
CROSSREFS
Column k=2 of A238349 and of A238350.
Sequence in context: A243391 A000206 A368031 * A075223 A071332 A006791
KEYWORD
nonn
AUTHOR
Joerg Arndt and Alois P. Heinz, Apr 11 2014
STATUS
approved