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
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..403
Programs
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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 *)
Formula
a(n) ~ c * 2^n * n! / n^(1/4), where c = 1.054816768531988358301631965137203014379828345839423725829486842843413035459... - Vaclav Kotesovec, May 14 2020