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A191826
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Number of n-step three-sided prudent self-avoiding walks ending on the left side of their box.
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6
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1, 3, 7, 18, 45, 114, 286, 721, 1812, 4558, 11449, 28757, 72172, 181074, 454043, 1138082, 2851369, 7141220, 17878222, 44743249, 111939780, 279966725, 700000634, 1749720020, 4372423409, 10923598469, 27283760095, 68130718351, 170093054202
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listen;
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OFFSET
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0,2
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LINKS
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EXAMPLE
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a(3) = 18: ENW, NNN, NNW, NWN, NWW, NWS, WNN, WNW, WWN, WWW, WWS, WSW, WSS, SWN, SWW, SWS, SSW, SSS.
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MAPLE
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b:= proc(d, i, n, x, y, w) option remember;
`if`(w>n, 0, `if`(n=0, `if`(w=0, 1, 0),
`if`(d in [0, 1] or d<>3 and x=0 or d=2 and i,
b(1, evalb(x=0), n-1, max(x-1, 0), y, w+1), 0) +
`if`(d in [0, 2] or d<>4 and (y=0 or i),
b(2, evalb(y=0), n-1, x, max(y-1, 0), w), 0) +
`if`(d in [0, 3] or d<>1 and w=0 or d=2 and i,
b(3, evalb(w=0), n-1, x+1, y, max(w-1, 0)), 0) +
`if`(d in [0, 4] or d<>2 and i,
b(4, false, n-1, x, y+1, w), 0)))
end:
a:= n-> b(0, true, n, 0, 0, 0):
seq(a(n), n=0..30);
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MATHEMATICA
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b[d_, i_, n_, x_, y_, w_] := b[d, i, n, x, y, w] = If[w > n, 0, If[n == 0, If[w == 0, 1, 0], If[MatchQ[d, 0|1] || d != 3 && x == 0 || d == 2 && i, b[1, x == 0, n-1, Max[x-1, 0], y, w+1], 0] + If[MatchQ[d, 0|2] || d != 4 && (y == 0 || i), b[2, y == 0, n-1, x, Max[y-1, 0], w], 0]+ If[MatchQ[d, 0|3] || d != 1 && w == 0 || d == 2 && i, b[3, w == 0, n-1, x+1, y, Max[w-1, 0]], 0] + If[MatchQ[d, 0|4] || d != 2 && i, b[4, s == 0, n-1, x, y+1, w], 0]]]; a[n_] := b[0, True, n, 0, 0, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 22 2014, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn,walk
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AUTHOR
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STATUS
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approved
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