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

0,2

Links

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

Formula

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

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 *)

Crossrefs

Cf. A317985.

Sequence in context: A167137 A279434 A000556 * A125598 A267436 A294215

Adjacent sequences: A320509 A320510 A320511 * A320513 A320514 A320515

Keyword

nonn,walk

Author

Alois P. Heinz, Oct 22 2018

Status

approved