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 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). 2

%I

%S 1,4,21,148,980,6444,41888,270088,1730079,11023480,69930146,441988260,

%T 2784820519,17499028820,109701885600,686313858480,4285914086100,

%U 26721615383496,166361793070466,1034375862301240,6423778211164860,39850734775066644,246976735839649218

%N 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).

%H Alois P. Heinz, <a href="/A327872/b327872.txt">Table of n, a(n) for n = 0..1276</a>

%H Alois P. Heinz, <a href="/A327871/a327871.gif">Animation of A327871(5) = 369 walks with a(5) = 6444 nodes</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(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

%p b:= proc(x, y, t) option remember; (p-> p+[0, p[1]])(`if`(

%p min(x, y)<0, 0, `if`(max(x, y)=0, [1, 0], 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)[2]:

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

%Y Cf. A327871.

%K nonn,walk

%O 0,2

%A _Alois P. Heinz_, Sep 28 2019

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Last modified December 10 04:15 EST 2019. Contains 329885 sequences. (Running on oeis4.)