The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A327871

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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).
(history; published version)

#24 by Susanna Cuyler at Wed May 19 10:25:03 EDT 2021

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proposed

approved

#23 by Peter Luschny at Wed May 19 08:34:56 EDT 2021

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editing

proposed

#22 by Peter Luschny at Wed May 19 08:33:05 EDT 2021

FORMULA

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) for n >= 1. - _Peter Luschny_, May 19 2021(End)

CROSSREFS

Cf. A001006, A026300, A086246, A327872.

#21 by Peter Luschny at Wed May 19 05:06:37 EDT 2021

FORMULA

a(n) = binomial(2*n - 1, n - 1)*hypergeom([(2 - n)/2, (1 - n)/2], [n + 2], 4) for n >= 1. - Peter Luschny, May 19 2021

MATHEMATICA

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 *)

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approved

editing

#20 by Alois P. Heinz at Wed May 13 05:23:27 EDT 2020

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proposed

approved

#19 by Jean-François Alcover at Wed May 13 04:31:52 EDT 2020

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editing

proposed

#18 by Jean-François Alcover at Wed May 13 04:31:48 EDT 2020

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 *)

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approved

editing

#17 by Vaclav Kotesovec at Sat Oct 12 10:30:03 EDT 2019

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editing

approved

#16 by Vaclav Kotesovec at Sat Oct 12 10:28:41 EDT 2019

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

STATUS

approved

editing

#15 by Susanna Cuyler at Wed Oct 09 13:36:21 EDT 2019

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proposed

approved