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).

1, 2, 7, 38, 284, 2691, 30890, 416449, 6448243, 112751661, 2197200541, 47214026822, 1109022356759, 28269085769331, 777140210643254, 22918982645377342, 721764216387297451, 24173661551378798838, 857993099925433301350, 32168967331652245055171

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-François 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