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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). 2
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 (list; graph; refs; listen; history; text; internal format)
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

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-Fran├žois Alcover, May 13 2020, after Maple *)

CROSSREFS

Cf. A001006, 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

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Last modified May 17 18:17 EDT 2021. Contains 343986 sequences. (Running on oeis4.)