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A327872 Total number of nodes in all self-avoiding planar walks starting at (0,0), ending at (n,n), remaining in the first quadrant and using steps (0,1), (-1,1), and (1,-1) with the restriction that (-1,1) and (1,-1) are always immediately followed by (0,1).

%I A327872 #17 May 13 2020 05:21:50
%S A327872 1,4,21,148,980,6444,41888,270088,1730079,11023480,69930146,441988260,
%T A327872 2784820519,17499028820,109701885600,686313858480,4285914086100,
%U A327872 26721615383496,166361793070466,1034375862301240,6423778211164860,39850734775066644,246976735839649218
%N A327872 Total number of nodes in all self-avoiding planar walks starting at (0,0), ending at (n,n), remaining in the first quadrant and using steps (0,1), (-1,1), and (1,-1) with the restriction that (-1,1) and (1,-1) are always immediately followed by (0,1).
%H A327872 Alois P. Heinz, <a href="/A327872/b327872.txt">Table of n, a(n) for n = 0..1276</a>
%H A327872 Alois P. Heinz, <a href="/A327871/a327871.gif">Animation of A327871(5) = 369 walks with a(5) = 6444 nodes</a>
%H A327872 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%H A327872 Wikipedia, <a href="https://en.wikipedia.org/wiki/Self-avoiding_walk">Self-avoiding walk</a>
%F A327872 a(n) ~ sqrt(113 - 179/sqrt(13)) * (70 + 26*sqrt(13))^n * sqrt(n) / (sqrt(Pi) * 2^(3/2) * 3^(3*n + 3/2)). - _Vaclav Kotesovec_, Oct 12 2019
%p A327872 b:= proc(x, y, t) option remember; (p-> p+[0, p[1]])(`if`(
%p A327872        min(x, y)<0, 0, `if`(max(x, y)=0, [1, 0], b(x-1, y, 1)+
%p A327872       `if`(t=1, b(x-1, y+1, 0)+b(x+1, y-1, 0), 0))))
%p A327872     end:
%p A327872 a:= n-> b(n$2, 0)[2]:
%p A327872 seq(a(n), n=0..25);
%t A327872 b[x_, y_, t_] := b[x, y, t] = Function[p, p + {0, p[[1]]}][If[Min[x, y] < 0, {0, 0}, If[Max[x, y] == 0, {1, 0}, b[x - 1, y, 1] + If[t == 1, b[x - 1, y + 1, 0] + b[x + 1, y - 1, 0], 0]]]];
%t A327872 a[n_] := b[n, n, 0][[2]];
%t A327872 a /@ Range[0, 25] (* _Jean-François Alcover_, May 13 2020, after Maple *)
%Y A327872 Cf. A327871.
%K A327872 nonn,walk
%O A327872 0,2
%A A327872 _Alois P. Heinz_, Sep 28 2019