A303478 Total number of nodes summed over all 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) with the restrictions that (0,1) and (1,-1) are never used above the diagonal and (1,0) and (-1,1) are never used below the diagonal and (1,1) can only be used below the diagonal.

1, 2, 8, 26, 114, 439, 1932, 8100, 35943, 157300, 705592, 3160856, 14328020, 65101759, 297739250, 1365807037, 6291534200, 29062368336, 134648511026, 625312014415, 2910626302818, 13574639952187, 63425918284235, 296834615623188, 1391290575835806

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1465

Alois P. Heinz, Animation of A303477(6)=186 walks with a(6)=1932 nodes

Wikipedia, Lattice path

Wikipedia, Self-avoiding walk

Maple

b:= proc(x, y) option remember; (p-> p+[0, p[1]])(
      `if`(x<0 or y<0, 0, `if`(x=0 and y=0, [1, 0],
      `if`(x>y-2, b(x, y-1), 0)+`if`(x<y+2, b(x-1, y), 0)+
      `if`(x>y, b(x-1, y-1), 0)+`if`(x<y-1, b(x+1, y-1), 0)+
      `if`(x>y+1, b(x-1, y+1), 0))))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..30);

Mathematica

b[x_, y_] := b[x, y] = Function[p, p + {0, p[[1]]}][
  If[x < 0 || y < 0, {0, 0}, If[x == 0 && y == 0, {1, 0},
  If[x > y - 2, b[x, y - 1], {0, 0}] + If[x < y + 2, b[x - 1, y], 0] +
  If[x > y, b[x - 1, y - 1], 0] + If[x < y - 1, b[x + 1, y - 1], 0] +
  If[x > y + 1, b[x - 1, y + 1], 0]]]];
a[n_] := b[n, 0][[2]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Crossrefs

Cf. A303477.

Sequence in context: A150680 A150681 A150682 * A150683 A150684 A150685

Adjacent sequences: A303475 A303476 A303477 * A303479 A303480 A303481

Keyword

nonn,walk

Author

Alois P. Heinz, Apr 24 2018

Status

approved