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A272763
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Number of n-step self-avoiding walks on the square lattice with diagonals allowed (Moore neighborhood).
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6
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1, 8, 56, 368, 2336, 14576, 89928, 550504, 3349864, 20290360, 122445504, 736685008, 4421048016, 26475370088, 158257613848, 944493430152, 5628996811904
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OFFSET
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0,2
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COMMENTS
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Moore neighborhood :
o o o
o x o
o o o
Von Neumann neighborhood (A001411):
o
o x o
o
Note that the path avoids already visited lattice points, but can intersect itself (two diagonal steps). A nonintersecting version is A272773.
The Moore neighborhood characterizes king tours. # Rainer Rosenthal, Jan 05 2019
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LINKS
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MAPLE
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# For starting point stp and list Ldir of n directions (1..8)
# construct the points of the whole path and count them.
# If there are n+1 then the path is self-avoiding.
isSelfAvoiding := proc(Ldir) local Delta, dir, ep, path;
Delta := [[1, 0], [1, 1], [0, 1], [-1, 1], [-1, 0], [-1, -1], [0, -1], [1, -1]];
ep := [0, 0]; path := {ep};
for dir in Ldir do
ep := ep + Delta[dir];
path := {op(path), ep};
od;
return evalb(nops(path)=nops(Ldir)+1);
end:
# Count only king tours which are self-avoiding
A272763 := proc(n) local count, T, p;
count := 0:
T := combinat[cartprod]([seq([$1..8], j=1..n)]):
while not T[finished] do
p := T[nextvalue]();
if isSelfAvoiding(p) then count := count+1; fi;
od:
return count;
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MATHEMATICA
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mo=Most@Tuples[{-1, 1, 0}, 2]; a[0]=1; a[tg_, p_: {{0, 0}}] := Block[{e, mv = Complement[Last[p] + # & /@ mo, p]}, If[tg == 1, Length@mv, Sum[a[tg - 1, Append[p, e]], {e, mv}]]]; a /@ Range[0, 7] (* Giovanni Resta, May 06 2016 *)
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CROSSREFS
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KEYWORD
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nonn,walk,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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