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A191825
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Number of n-step three-sided prudent self-avoiding walks ending on the bottom side of their box.
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6
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1, 3, 7, 17, 39, 91, 209, 487, 1135, 2663, 6259, 14769, 34917, 82763, 196489, 467325, 1112941, 2654015, 6335751, 15140525, 36212889, 86685101, 207653983, 497772203, 1193946381, 2865399103, 6880328883, 16528823695, 39725234877, 95514665625
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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) = 17: EEE, EES, ESE, ESS, NES, NWS, WWW, WWS, WSW, WSS, SEE, SES, SWW, SWS, SSE, SSW, SSS.
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MAPLE
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b:= proc(d, i, n, x, y, w, s)
if w>x then b([3, 2, 1, 4][d], i, n, w, y, x, s)
else b(d, i, n, x, y, w, s):=
`if`(s>n, 0, `if`(n=0, `if`(s=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, s), 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, s+1), 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), s), 0) +
`if`(d in [0, 4] or d<>2 and (s=0 or i),
b(4, evalb(s=0), n-1, x, y+1, w, max(s-1, 0)), 0)))
fi
end:
a:= n-> b(0, true, n, 0, 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_, s_] := b[d, i, n, x, y, w, s] = If[w>x, b[{3, 2, 1, 4}[[d]], i, n, w, y, x, s], If[s>n, 0, If[n == 0, If[s == 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, s], 0]+If[MatchQ[d, 0|2] || d != 4 && (y == 0 || i), b[2, y == 0, n-1, x, Max[y-1, 0], w, s+1], 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], s], 0]+If[MatchQ[d, 0|4] || d != 2 && (s == 0 || i), b[4, s == 0, n-1, x, y+1, w, Max[s-1, 0]], 0]]]]; a[n_] := b[0, True, n, 0, 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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