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A128761 Number of compositions of n with parts in N which avoid the consecutive pattern 123.

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