A238351 Number of compositions p(1)+p(2)+...+p(k) = n such that for no part p(i) = i (compositions without fixed points).

1, 0, 1, 2, 3, 6, 11, 22, 42, 82, 161, 316, 624, 1235, 2449, 4864, 9676, 19267, 38399, 76582, 152819, 305085, 609282, 1217140, 2431992, 4860306, 9714696, 19419870, 38824406, 77624110, 155208405, 310352615, 620601689, 1241036325, 2481803050, 4963170896

Offset

0,4

Comments

Column k=0 of A238349 and of A238350.

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 = 0..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 = A048651/2 = 0.14439404754330121... - Vaclav Kotesovec, May 01 2014

Example

The a(7) = 22 such compositions are:
01:  [ 2 1 1 1 1 1 ]
02:  [ 2 1 1 1 2 ]
03:  [ 2 1 1 2 1 ]
04:  [ 2 1 1 3 ]
05:  [ 2 1 2 1 1 ]
06:  [ 2 1 2 2 ]
07:  [ 2 1 4 ]
08:  [ 2 3 1 1 ]
09:  [ 2 3 2 ]
10:  [ 2 4 1 ]
11:  [ 2 5 ]
12:  [ 3 1 1 1 1 ]
13:  [ 3 1 1 2 ]
14:  [ 3 1 2 1 ]
15:  [ 3 3 1 ]
16:  [ 3 4 ]
17:  [ 4 1 1 1 ]
18:  [ 4 1 2 ]
19:  [ 4 3 ]
20:  [ 5 1 1 ]
21:  [ 6 1 ]
22:  [ 7 ]

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
       add(`if`(i=j, 0, b(n-j, i+1)), j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..50);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[If[i == j, 0, b[n-j, i+1]], {j, 1, n}]]; a[n_] := b[n, 1]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 06 2014, after Maple *)

Crossrefs

Sequence in context: A036589 A251656 A123341 * A043328 A141072 A002083

Adjacent sequences: A238348 A238349 A238350 * A238352 A238353 A238354

Keyword

nonn

Author

Joerg Arndt and Alois P. Heinz, Feb 25 2014

Status

approved