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A317985 Number of self-avoiding planar walks starting at (0,0), ending at (n,0), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1) such that (0,1) is never used directly before or after (1,0) or (1,1). 3
1, 2, 7, 38, 284, 2691, 30890, 416449, 6448243, 112751661, 2197200541, 47214026822, 1109022356759, 28269085769331, 777140210643254, 22918982645377342, 721764216387297451, 24173661551378798838, 857993099925433301350, 32168967331652245055171 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..403

FORMULA

a(n) ~ c * 2^n * n! / n^(1/4), where c = 1.054816768531988358301631965137203014379828345839423725829486842843413035459... - Vaclav Kotesovec, May 14 2020

MAPLE

a:= proc(n) option remember; `if`(n<4, [1, 2, 7, 38][n+1],

      2*n*a(n-1) -(n-2)*a(n-2) -(2*n-5)*a(n-3))

    end:

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

MATHEMATICA

a = DifferenceRoot[Function[{y, n}, {(2n+1) y[n] + (n+1) y[n+1] + (-2n-6)* y[n+2] + y[n+3] == 0, y[0] == 1, y[1] == 2, y[2] == 7, y[3] == 38}]];

Table[a[n], {n, 0, 25}] (* Jean-Fran├žois Alcover, May 12 2020, after Maple *)

nmax = 20; CoefficientList[Simplify[Normal[Series[-1 - 1/x^(3/4) * E^(-1/(2*x) + (3*ArcTanh[(1 + 4*x)/Sqrt[17]])/(4*Sqrt[17]))* (-2 + x + 2*x^2)^(1/8) * Integrate[E^(1/(2*x)) * Simplify[Normal[Series[(-2 + 2*x + x^2)/(x^(5/4)*(-2 + x + 2*x^2)^(9/8))/ E^(3*ArcTanh[(1 + 4*x)/Sqrt[17]] / (4*Sqrt[17])), {x, 0, nmax}]], x > 0], x], {x, 0, nmax}]], x > 0], x] (* Vaclav Kotesovec, May 14 2020 *)

CROSSREFS

Cf. A277358, A320512.

Sequence in context: A088792 A114160 A145159 * A084552 A094664 A001858

Adjacent sequences:  A317982 A317983 A317984 * A317986 A317987 A317988

KEYWORD

nonn,walk

AUTHOR

Alois P. Heinz, Oct 02 2018

STATUS

approved

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Last modified May 20 01:54 EDT 2022. Contains 353849 sequences. (Running on oeis4.)