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A238351
Number of compositions p(1)+p(2)+...+p(k) = n such that for no part p(i) = i (compositions without fixed points).
25
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: A251656 A123341 A372989 * A043328 A141072 A002083
KEYWORD
nonn
AUTHOR
Joerg Arndt and Alois P. Heinz, Feb 25 2014
STATUS
approved