%I #13 Jun 28 2024 10:07:44

%S 1,3,8,20,50,126,322,834,2187,5797,15510,41834,113633,310571,853466,

%T 2356778,6536381,18199283,50852018,142547558,400763222,1129760414,

%U 3192727796,9043402500,25669818475,73007772801,208023278208

%N Number of possible column states for self-avoiding polygons in a slit of width n.

%C Number of Dyck (n+1)-paths whose maximum ascent length is 2. - _David Scambler_, Aug 22 2012

%H J. Alvarez, E.J. Janse van Rensburg et al. <a href="http://www.math.yorku.ca/Who/Faculty/Rensburg/Preprints/slits.pdf">Self-avoiding walks and polygons in slits</a>.

%H Louis Marin, <a href="https://arxiv.org/abs/2406.16413">Counting Polyominoes in a Rectangle b X h</a>, arXiv:2406.16413 [cs.DM], 2024. See p. 148.

%F a(n) = Sum_{m=1..[(n+1)/2]} (n+1)!/((n+1-2m)!m!(m+1)!).

%F a(n) = A001006(n+2)-1.

%F D-finite with recurrence (n+3)*a(n) +(-4*n-7)*a(n-1) +(2*n+3)*a(n-2) +(4*n-5) *a(n-3) +3*(-n+2)*a(n-4)=0. - _R. J. Mathar_, Nov 01 2021

%Y Cf. A055151, A080159, A088617, A107131, A097610.

%Y Column k=2 of A203717 (shifted).

%K easy,nonn

%O 1,2

%A _R. J. Mathar_, Jul 11 2008