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A107069
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Number of self-avoiding walks of length n on an infinite triangular prism starting at the origin.
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0
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1, 4, 12, 34, 90, 222, 542, 1302, 3058, 7186, 16714, 38670, 89358, 205710, 472906, 1086138, 2491666, 5713318, 13094950, 30003190, 68731010, 157423986, 360530346, 825626942, 1890615518, 4329196974, 9912914314, 22698017834, 51972012258, 119000208806
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OFFSET
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0,2
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COMMENTS
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The discrete space in which the walking happens is a triangular prism infinite in both directions along the x-axis. One vertex is the root, the origin. The basis is the set of single-step vectors, which we abbreviate as l (left), r (right), c (one step "clockwise" around the triangle) and c- (one step counterclockwise, more properly denoted c^-1).
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LINKS
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EXAMPLE
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a(0) = 1, as there is one self-avoiding walk of length 0, namely the null-walk (the walk whose steps are the null set).
a(1) = 4 because (using the terminology in the Comment), the 4 possible 1-step walks are W_1 = {l,r,c,c-}.
a(2) = 12 because the set of legal 2-step walks are {l^2, lc, lc-, r^2, rc, rc-, c^2, cl, cr, c^-2, c-l, c-r}.
a(3) = 34 because we have every W_2 concatenated with {l,r,c,c-} except for those with immediate violations (lr etc.) and those two which go in a triangle {c^3, c^-3}; hence a(3) = 3*a(2) - 2 = 3*12 - 2 = 36 - 2 = 34.
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PROG
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(Python)
w = [[[(0, 0)]]]
for n in range(1, 15):
nw = []
for walk in w[-1]:
(x, t) = walk[-1]
nss = [(x-1, t), (x+1, t), (x, (t+1)%3), (x, (t-1)%3)]
for ns in nss:
if ns not in walk:
nw.append(walk[:] + [ns])
w.append(nw)
print([len(x) for x in w])
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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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EXTENSIONS
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STATUS
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approved
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