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%I #10 Nov 07 2014 03:44:15
%S 1,1,3,7,12,30,61,126,258,537,1083,2205,4465,9023,18192,36612,73633,
%T 147893,296818,595313,1193351,2391121,4789448,9590503,19199906,
%U 38430421,76910470,153901337,307932963,616076971,1232495756,2465545205,4931986957,9865425657
%N Number of compositions of n having exactly three fixed points.
%H Joerg Arndt and Alois P. Heinz, <a href="/A240738/b240738.txt">Table of n, a(n) for n = 6..1000</a>
%F a(n) ~ c * 2^n, where c = 0.01795631780689407343024911217251418606332716557572090051127381129853009022... . - _Vaclav Kotesovec_, Sep 07 2014
%e a(8) = 3: 1214, 1232, 12311.
%e a(9) = 7: 1134, 1224, 1233, 12141, 12312, 12321, 123111.
%e a(10) = 12: 11341, 12115, 12142, 12241, 12313, 12322, 12331, 121411, 123112, 123121, 123211, 1231111.
%p b:= proc(n, i) option remember; `if`(n=0, 1, series(
%p add(b(n-j, i+1)*`if`(i=j, x, 1), j=1..n), x, 4))
%p end:
%p a:= n-> coeff(b(n, 1), x, 3):
%p seq(a(n), n=6..45);
%t 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, 4}]]; a[n_] := SeriesCoefficient[b[n, 1], {x, 0, 3}]; Table[a[n], {n, 6, 45}] (* _Jean-François Alcover_, Nov 07 2014, after Maple *)
%Y Column k=3 of A238349 and of A238350.
%K nonn
%O 6,3
%A _Joerg Arndt_ and _Alois P. Heinz_, Apr 11 2014
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