A190589 Number of two-sided n-step prudent walks avoiding two or more consecutive west steps and two or more consecutive south steps.

1, 4, 8, 20, 46, 110, 260, 616, 1456, 3442, 8134, 19218, 45394, 107204, 253140, 597666, 1410948, 3330622, 7861542, 18555092, 43792062, 103349390, 243895352, 575552060, 1358167086, 3204874310, 7562397040, 17844314546, 42104959766, 99348325696, 234413460808

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..800

Shanzhen Gao, Keh-Hsun Chen, Tackling Sequences From Prudent Self-Avoiding Walks, FCS'14, The 2014 International Conference on Foundations of Computer Science.

Formula

G.f.: 1/((1-2*t-t^2+t^3)*(1-2*t-2*t^3))*(t*(1-t)^2*sqrt((1-t-t^3)^2-4*t^4)+1-t-2*t^2-2*t^4+4*t^5-t^6), see sequence 13 in link. - Michel Marcus, May 06 2015

Maple

b:= proc(d, i, n, x, y) option remember;
      `if`(n=0, 1,
         `if`(d<>3, b(1, x=0, n-1, max(x-1, 0), y), 0) +
         `if`(d<>4, b(2, y=0, n-1, x, max(y-1, 0)), 0) +
         `if`(d=0 or d=2 and i, b(3, false, n-1, x+1, y), 0) +
         `if`(d=0 or d=1 and i, b(4, false, n-1, x, y+1), 0))
    end:
a:= n-> b(0, false, n, 0, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 09 2011

Mathematica

b[d_, i_, n_, x_, y_] := b[d, i, n, x, y] =
If[n == 0, 1,
  If[d != 3, b[1, x == 0, n - 1, Max[x - 1, 0], y], 0] +
  If[d != 4, b[2, y == 0, n - 1, x, Max[y - 1, 0]], 0] +
  If[d == 0 || d == 2 && i, b[3, False, n - 1, x + 1, y], 0] +
  If[d == 0 || d == 1 && i, b[4, False, n - 1, x, y + 1], 0]
];
a[n_] := b[0, False, n, 0, 0];
a /@ Range[0, 30] (* Jean-François Alcover, May 13 2020, after Alois P. Heinz *)

Crossrefs

Equals 2*A190586 - A190587.

Sequence in context: A334706 A095804 A084219 * A009916 A203167 A123861

Adjacent sequences: A190586 A190587 A190588 * A190590 A190591 A190592

Keyword

nonn,walk

Author

Shanzhen Gao, May 13 2011

Extensions

More terms from Alois P. Heinz, Jun 09 2011

Status

approved