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%I #21 Mar 03 2021 21:50:35
%S 1,0,1,2,3,6,11,22,42,82,161,316,624,1235,2449,4864,9676,19267,38399,
%T 76582,152819,305085,609282,1217140,2431992,4860306,9714696,19419870,
%U 38824406,77624110,155208405,310352615,620601689,1241036325,2481803050,4963170896
%N Number of compositions p(1)+p(2)+...+p(k) = n such that for no part p(i) = i (compositions without fixed points).
%C Column k=0 of A238349 and of A238350.
%D M. Archibald, A. Blecher and A. Knopfmacher, Fixed points in compositions and words, accepted by the Journal of Integer Sequences
%H Joerg Arndt and Alois P. Heinz, <a href="/A238351/b238351.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Archibald, A. Blecher, and A. Knopfmacher, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Blecher/arch14.html">Fixed Points in Compositions and Words</a>, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.
%F a(n) ~ c * 2^n, where c = A048651/2 = 0.14439404754330121... - _Vaclav Kotesovec_, May 01 2014
%e The a(7) = 22 such compositions are:
%e 01: [ 2 1 1 1 1 1 ]
%e 02: [ 2 1 1 1 2 ]
%e 03: [ 2 1 1 2 1 ]
%e 04: [ 2 1 1 3 ]
%e 05: [ 2 1 2 1 1 ]
%e 06: [ 2 1 2 2 ]
%e 07: [ 2 1 4 ]
%e 08: [ 2 3 1 1 ]
%e 09: [ 2 3 2 ]
%e 10: [ 2 4 1 ]
%e 11: [ 2 5 ]
%e 12: [ 3 1 1 1 1 ]
%e 13: [ 3 1 1 2 ]
%e 14: [ 3 1 2 1 ]
%e 15: [ 3 3 1 ]
%e 16: [ 3 4 ]
%e 17: [ 4 1 1 1 ]
%e 18: [ 4 1 2 ]
%e 19: [ 4 3 ]
%e 20: [ 5 1 1 ]
%e 21: [ 6 1 ]
%e 22: [ 7 ]
%p b:= proc(n, i) option remember; `if`(n=0, 1,
%p add(`if`(i=j, 0, b(n-j, i+1)), j=1..n))
%p end:
%p a:= n-> b(n, 1):
%p seq(a(n), n=0..50);
%t 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 *)
%K nonn
%O 0,4
%A _Joerg Arndt_ and _Alois P. Heinz_, Feb 25 2014
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