|
%I #17 Sep 20 2019 03:08:39
%S 1,1,2,4,8,16,31,61,119,232,453,883,1721,3354,6536,12735,24813,48344,
%T 94189,183506,357518,696534,1357019,2643798,5150746,10034865,19550268,
%U 38088486,74205248,144569092,281654211,548727863,1069049370,2082756500
%N Number of compositions of n with parts in N which avoid the consecutive pattern 123.
%H Alois P. Heinz, <a href="/A128761/b128761.txt">Table of n, a(n) for n = 0..1000</a>
%H S. Heubach and T. Mansour, <a href="https://arxiv.org/abs/math/0603285">Enumeration of 3-letter patterns in combinations</a>, arXiv:math/0603285 [math.CO], 2006.
%F The Heubach/Mansour paper has a complicated g.f.
%F a(n) ~ c * d^n, where d = 1.948232199887283224240693518762976752988220177086321158242512704029011807341..., c = 0.57609601848694597639954632728322472031509789101742496394456882851645843... - _Vaclav Kotesovec_, Sep 20 2019
%p b:= proc(n, t, l) option remember; `if`(n=0, 1, add(
%p b(n-j, is(j>l), j), j=1..min(n, `if`(t, l, n))))
%p end:
%p a:= n-> b(n, false, n):
%p seq(a(n), n=0..35); # _Alois P. Heinz_, Oct 24 2017
%t b[n_, t_, l_] := b[n, t, l] = If[n == 0, 1, Sum[b[n - j, j > l, j], {j, 1, Min[n, If[t, l, n]]}]];
%t a[n_] := b[n, False, n];
%t Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Nov 14 2017, after _Alois P. Heinz_ *)
%K nonn
%O 0,3
%A _Ralf Stephan_, May 08 2007
%E More terms from _Vladeta Jovovic_, Oct 03 2007
|
