A328140 Total number of nodes in all self-avoiding planar walks starting at (0,0), ending at (n,0), not extending above the line (x,2x) or below the line (x,-2x), 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).

1, 2, 8, 22, 81, 260, 854, 2738, 8710, 27550, 86696, 271726, 848681, 2642662, 8207726, 25434686, 78663773, 242865100, 748650655, 2304552576, 7085109570, 21757557324, 66745430284, 204559349854, 626379255199, 1916485240548, 5859359429380, 17901726707224

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..2093

Alois P. Heinz, Animation of A328139(7) = 222 walks with a(7) = 2738 nodes

Wikipedia, Lattice path

Wikipedia, Self-avoiding walk

Maple

b:= proc(x, y, t) option remember; (p-> p+[0, p[1]])(`if`(x<0
       or abs(y)>2*x, 0, `if`(x=0, [1, 0], b(x-1, y, 1)+
      `if`(t=1, b(x-1, y+1, 0)+b(x+1, y-1, 0), 0))))
    end:
a:= n-> b(n, 0$2)[2]:
seq(a(n), n=0..32);

Mathematica

b[x_, y_, t_] := b[x, y, t] = Function[p, p + {0, p[[1]]}][If[x < 0 || Abs[y] > 2x, {0, 0}, If[x == 0, {1, 0}, 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, 0, 0][[2]];
a /@ Range[0, 32] (* Jean-François Alcover, May 13 2020, after Maple *)

Crossrefs

Cf. A328139.

Sequence in context: A121135 A183410 A072929 * A053958 A300370 A191643

Adjacent sequences: A328137 A328138 A328139 * A328141 A328142 A328143

Keyword

nonn,walk

Author

Alois P. Heinz, Oct 04 2019

Status

approved