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).

1, 4, 21, 148, 980, 6444, 41888, 270088, 1730079, 11023480, 69930146, 441988260, 2784820519, 17499028820, 109701885600, 686313858480, 4285914086100, 26721615383496, 166361793070466, 1034375862301240, 6423778211164860, 39850734775066644, 246976735839649218

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1276

Alois P. Heinz, Animation of A327871(5) = 369 walks with a(5) = 6444 nodes

Wikipedia, Lattice path

Wikipedia, Self-avoiding walk

Formula

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

Maple

b:= proc(x, y, t) option remember; (p-> p+[0, p[1]])(`if`(
       min(x, y)<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))))
    end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..25);

Mathematica

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]]]];
a[n_] := b[n, n, 0][[2]];
a /@ Range[0, 25] (* Jean-François Alcover, May 13 2020, after Maple *)

Crossrefs

Cf. A327871.

Sequence in context: A308337 A307525 A345749 * A163861 A247054 A006153

Adjacent sequences: A327869 A327870 A327871 * A327873 A327874 A327875

Keyword

nonn,walk

Author

Alois P. Heinz, Sep 28 2019

Status

approved