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A191828
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Number of n-step three-sided prudent self-avoiding walks ending at the northwest corner of their box.
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6
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1, 2, 4, 10, 24, 59, 143, 351, 860, 2117, 5211, 12856, 31734, 78431, 193951, 479983, 1188388, 2943786, 7294704, 18082477, 44835711, 111197870, 275839085, 684372592, 1698221877, 4214570553, 10460699937, 25966317723, 64460852039
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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(4) = 24: ENNW, ENWN, ENWW, NENW, NNNN, NNNW, NNWN, NNWW, NWNN, NWNW, NWWN, NWWW, WNNN, WNNW, WNWN, WNWW, WWNN, WWNW, WWWN, WWWW, WSWN, SWNN, SWNW, SWWN.
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MAPLE
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b:= proc(d, i, n, x, y, w) option remember;
`if`(y+w>n, 0, `if`(n=0, `if`(y=0 and w=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+w>n, 0, If[n == 0, If[y == 0 && 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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