A327871 Number of 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, 1, 3, 14, 70, 369, 2002, 11076, 62127, 352070, 2010998, 11559030, 66780155, 387444085, 2255875650, 13174629240, 77143234950, 452738296890, 2662359410158, 15683996769460, 92540962166016, 546799192200261, 3235027635603828, 19161631961190036, 113617798289197650

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1280

Alois P. Heinz, Animation of a(5) = 369 walks

Wikipedia, Lattice path

Wikipedia, Self-avoiding walk

Formula

a(n) ~ sqrt(5 + 1/sqrt(13)) * (70 + 26*sqrt(13))^n / (2^(3/2) * sqrt(Pi*n) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Oct 12 2019

From Peter Luschny, May 19 2021: (Start)

Next three formulas for n >= 1:

a(n) = A026300(2*n - 1, n - 1).

a(n) = Sum_{j=0..floor((n-1)/2)} C(2*n-1, 2*j + n)*(C(2*j + n, j) - C(2*j +n, j-1)).

a(n) = binomial(2*n - 1, n - 1)*hypergeom([(2 - n)/2, (1 - n)/2], [n + 2], 4). (End)

Maple

b:= proc(x, y, t) option remember; `if`(min(x, y)<0, 0,
      `if`(max(x, y)=0, 1, 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):
seq(a(n), n=0..25);

Mathematica

b[x_, y_, t_] := b[x, y, t] = If[Min[x, y] < 0, 0, If[Max[x, y]==0, 1, 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];
a /@ Range[0, 25] (* Jean-François Alcover, May 13 2020, after Maple *)
a[n_] := Binomial[2n - 1, n - 1] Hypergeometric2F1[(2 - n)/2, (1 - n)/2, n + 2, 4];
a[0] := 1; Table[a[n], {n, 0, 24}] (* Peter Luschny, May 19 2021 *)

Crossrefs

Cf. A001006, A026300, A086246, A327872.

Sequence in context: A161939 A270598 A001579 * A006772 A320421 A009020

Adjacent sequences: A327868 A327869 A327870 * A327872 A327873 A327874

Keyword

nonn,walk

Author

Alois P. Heinz, Sep 28 2019

Status

approved