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A191824
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Number of n-step three-sided prudent self-avoiding walks ending on the top side of their box.
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6
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1, 3, 7, 19, 49, 127, 327, 841, 2157, 5523, 14115, 36019, 91779, 233557, 593625, 1507125, 3822395, 9685135, 24518103, 62016309, 156742115, 395865603, 999105203, 2519960799, 6352014717, 16002180569, 40291214021, 101395244925, 255042422247
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listen;
history;
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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) = 19: EEE, EEN, ENE, ENN, ENW, NEE, NEN, NNE, NNN, NNW, NWN, NWW, WNE, WNN, WNW, WWN, WWW, SEN, SWN.
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MAPLE
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b:= proc(d, i, n, x, y, w) option remember;
`if`(y>n, 0, `if`(n=0, `if`(y=0, 1, 0),
`if`(d in [0, 1] or d in [2, 4] 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 in [1, 3] 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 in [2, 4] 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 in [1, 3] 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[y>n, 0, If[n == 0, If[n == 0, 1, 0], If[MatchQ[d, 0|1] || MatchQ[d, 2|4] && x == 0 || d == 2 && i, b[1, x == 0, n-1, Max[x-1, 0], y, w+1], 0] + If[MatchQ[d, 0|2] || MatchQ[d, 1|3] && (y == 0 || i), b[2, y == 0, n-1, x, Max[y-1, 0], w], 0] + If[MatchQ[d, 0|3] || MatchQ[d, 2|4] && w == 0 || d == 2 && i, b[3, w == 0, n-1, x+1, y, Max[w-1, 0]], 0] + If[MatchQ[d, 0|4] || MatchQ[d, 1|3] && i, b[4, False, 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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