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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

%I

%S 1,1,3,14,70,369,2002,11076,62127,352070,2010998,11559030,66780155,

%T 387444085,2255875650,13174629240,77143234950,452738296890,

%U 2662359410158,15683996769460,92540962166016,546799192200261,3235027635603828,19161631961190036,113617798289197650

%N 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).

%H Alois P. Heinz, <a href="/A327871/b327871.txt">Table of n, a(n) for n = 0..1280</a>

%H Alois P. Heinz, <a href="/A327871/a327871.gif">Animation of a(5) = 369 walks</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Self-avoiding_walk">Self-avoiding walk</a>

%F 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

%p b:= proc(x, y, t) option remember; `if`(min(x, y)<0, 0,

%p `if`(max(x, y)=0, 1, b(x-1, y, 1)+

%p `if`(t=1, b(x-1, y+1, 0)+b(x+1, y-1, 0), 0)))

%p end:

%p a:= n-> b(n\$2, 0):

%p seq(a(n), n=0..25);

%Y Cf. A001006, A086246, A327872.

%K nonn,walk

%O 0,3

%A _Alois P. Heinz_, Sep 28 2019

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Last modified January 28 00:32 EST 2020. Contains 331313 sequences. (Running on oeis4.)