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A191652
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Number of n-step two-sided prudent self-avoiding walks ending on the top side of their box.
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3
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1, 3, 7, 18, 45, 113, 283, 709, 1775, 4442, 11111, 27781, 69433, 173468, 433229, 1081609, 2699521, 6735586, 16801355, 41898736, 104460505, 260378007, 648878481, 1616720044, 4027390409, 10030782405, 24978849433, 62192878443, 154825778335
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OFFSET
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0,2
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LINKS
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FORMULA
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EXAMPLE
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a(3) = 18: EEE, EEN, ENE, ENN, ENW, NEE, NEN, NNE, NNN, NNW, NWN, NWW, WNE, WNN, WNW, WWN, WWW, SEN.
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MAPLE
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b:= proc(d, i, n, x, y) option remember;
`if`(n=0, `if`(y=0, 1, 0),
`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 in [0, 3] or d=2 and i, b(3, false, n-1, x+1, y), 0) +
`if`(d in [0, 4] 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);
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MATHEMATICA
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b[d_, i_, n_, x_, y_] := b[d, i, n, x, y] = If [n == 0, If [y == 0, 1, 0], 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 == 3 || d == 2 && i, b[3, False, n - 1, x + 1, y], 0] + If[d == 0 || d == 4 || d == 1 && i, b[4, False, n - 1, x, y + 1], 0]];
a[n_] := b[0, False, n, 0, 0];
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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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