A128761 - OEIS

%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