A002903 -id:A002903 - cp's OEIS frontend

A001666 Number of n-step self-avoiding walks on b.c.c. lattice (version 2).

1, 8, 56, 392, 2648, 17960, 120056, 804824, 5351720, 35652680, 236291096, 1568049560, 10368669992, 68626647608, 453032542040, 2992783648424, 19731335857592, 130161040083608, 857282278813256, 5648892048530888, 37175039569217672, 244738250638121768

Offset: 0

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrey Zabolotskiy, Table of n, a(n) for n = 0..28 (from Schram et al.)
M. E. Fisher and M. F. Sykes, Excluded-volume problem and the Ising model of ferromagnetism, Phys. Rev. 114 (1959), 45-58.
A. J. Guttmann, On the critical behavior of self-avoiding walks II, J. Phys. A 22 (1989), 2807-2813. See Table 1.
Raoul D. Schram, Gerard T. Barkema, Rob H. Bisseling and Nathan Clisby, Exact enumeration of self-avoiding walks on BCC and FCC lattices, J. Stat. Mech. (2017) 083208; arXiv:1703.09340 [cond-mat.stat-mech], 2017. See Table I.
M. F. Sykes, Some counting theorems in the theory of the Ising problem and the excluded volume problem, J. Math. Phys., 2 (1961), 52-62.
M. F. Sykes et al., The asymptotic behavior of selfavoiding walks and returns on a lattice, J. Phys. A 5 (1972), 653-660.
Index entries for sequences related to b.c.c. lattice

Equals twice A002903 except for initial term.

Cf. A001336, A001412.

a(16)-a(18) from Bert Dobbelaere, Jan 16 2019
Terms a(19) and beyond from Schram et al. added by Andrey Zabolotskiy, Feb 02 2022
Edited by N. J. A. Sloane, Oct 16 2022

A107069 Number of self-avoiding walks of length n on an infinite triangular prism starting at the origin.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, May 10 2005

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).

			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.

Cf. A002902, A002903, A077817, A038577, A302408, A007825.

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])
# Andrey Zabolotskiy, Sep 19 2019

a(4) and a(5) corrected, a(6)-a(14) added by Andrey Zabolotskiy, Sep 19 2019

More terms from Andrey Zabolotskiy, Dec 04 2023

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A001666 Number of n-step self-avoiding walks on b.c.c. lattice (version 2).

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A107069 Number of self-avoiding walks of length n on an infinite triangular prism starting at the origin.

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