A320512 Total number of nodes summed over all self-avoiding planar walks starting at (0,0), ending at (n,0), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1) such that (0,1) is never used directly before or after (1,0) or (1,1).

1, 5, 31, 258, 2702, 33821, 492978, 8198218, 153136209, 3173544162, 72241986729, 1791612993205, 48074653669593, 1387590910289915, 42863756641047136, 1410904918289665343, 49296029555617568097, 1822020250023113834772, 71023629427964322798782

Offset: 0

Author

Alois P. Heinz, Oct 22 2018

Keywords

• nonn
• walk

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..402

Crossrefs

Cf. A317985.

Programs

Maple

b:= proc(x, y, i) option remember; (l-> `if`(min(x, y)<0, 0,
      `if`(max(x, y)=0, [1$2], add(`if`({i, j} in {{1, 2}, {3, 5},
       {4, 5}}, 0, (p-> p+[0, p[1]])(b(x-l[j][1], y-l[j][2], j))),
       j=1..5))))([[-1, 1], [1, -1], [1, 1], [1, 0], [0, 1]])
    end:
a:= n-> b(n, 0$2)[2]:
seq(a(n), n=0..20);

Mathematica

b[x_, y_, i_] := b[x, y, i] = With[{l = {{-1, 1}, {1, -1}, {1, 1}, {1, 0}, {0, 1}}}, If[Min[x, y] < 0, {0, 0}, If[Max[x, y] == 0, {1, 1}, Sum[If[ MemberQ[{{1, 2}, {3, 5}, {4, 5}}, Sort@{i, j}], {0, 0}, Function[p, p + {0, p[[1]]}][b[x - l[[j]][[1]], y - l[[j]][[2]], j]]], {j, 5}]]]];
a[n_] := b[n, 0, 0][[2]];
a /@ Range[0, 20] (* Jean-François Alcover, May 14 2020, after Maple *)

Formula

a(n) ~ c * n! * 2^n * n^(7/4), where c = 0.1758027947... - Vaclav Kotesovec, May 14 2020