A002902 -id:A002902 - cp's OEIS frontend

A001412 Number of n-step self-avoiding walks on cubic lattice.

1, 6, 30, 150, 726, 3534, 16926, 81390, 387966, 1853886, 8809878, 41934150, 198842742, 943974510, 4468911678, 21175146054, 100121875974, 473730252102, 2237723684094, 10576033219614, 49917327838734, 235710090502158, 1111781983442406, 5245988215191414, 24730180885580790, 116618841700433358, 549493796867100942, 2589874864863200574, 12198184788179866902, 57466913094951837030, 270569905525454674614

Offset: 0

N. J. A. Sloane

Steven R. Finch, Mathematical Constants, Cambridge, 2003, section 5.10, pp. 331-339.
B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 462.
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).

R. D. Schram, G. T. Barkema, and R. H. Bisseling, Table of n, a(n) for n = 0..36
N. Clisby, Enumerative combinatorics of lattice polymers, Notices AMS, 68:4 (2021), 504-515. (Excellent survey)
N. Clisby, R. Liang, and G. Slade, Self-avoiding walk enumeration via the lace expansion, J. Phys. A: Math. Theor. 40 (2007), 10973-11017, Table A5 for n<=30.
Steven R. Finch, Self-Avoiding-Walk Connective Constants
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, J. Phys. A 20 (1987), 1839-1854.
B. J. Hiley and M. F. Sykes, Probability of initial ring closure in the restricted random-walk model of a macromolecule, J. Chem. Phys., 34 (1961), 1531-1537.
D. S. McKenzie and C. Domb, The second osmotic virial coefficient of athermal polymer solutions, Proceedings of the Physical Society, 92 (1967) 632-649.
A. M. Nemirovsky, Karl F. Freed, Takao Ishinabe, and Jack F. Douglas, Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108.
D. Randall, Counting in Lattices: Combinatorial Problems from Statistical Mechanics, PhD Thesis (1994).
Raoul D. Schram, Gerard T. Barkema, and Rob H. Bisseling, Exact enumeration of self-avoiding walks, arXiv:1104.2184 [math-ph], 2011.
Nobu C. Shirai and Naoyuki Sakumichi, Negative Energetic Elasticity of Lattice Polymer Chain in Solvent, arXiv:2202.12483 [cond-mat.soft], 2022.
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, Self-avoiding walks on the simple cubic lattice, J. Chem. Phys., 39 (1963), 410-411.
M. F. Sykes, A. J. Guttmann, M. G. Watts, and P. D. Roberts, The asymptotic behavior of self-avoiding walks and returns on a lattice, J. Phys. A 5 (1972), 653-660.
M. F. Sykes, D. S. McKenzie, M. G. Watts, and J. L. Martin, The number of self-avoiding rings on a lattice, J. Phys. A 5 (1972), 661-666.

Cf. A002902, A078717, A001411, A001413.

mo = {{1, 0, 0}, {-1, 0, 0}, {0, 1, 0}, {0, -1, 0}, {0, 0, 1}, {0, 0, -1}}; a[0] = 1;
a[tg_, p_: {{0, 0, 0}}] := Block[{e, mv = Complement[Last[p] + # & /@ mo, p]},
If[tg == 1, Return[Length@mv],Sum[a[tg - 1, Append[p, e]], {e, mv}]]];
a /@ Range[0, 8]
(* Robert FERREOL, Nov 30 2018, after the program of Giovanni Resta in A001411 *)

def add(L,x):
    M=[y for y in L];M.append(x)
    return(M)
plus=lambda L,M : [x+y for x,y in zip(L,M)]
mo=[[1,0,0],[-1,0,0],[0,1,0],[0,-1,0],[0,0,1],[0,0,-1]]
def a(n,P=[[0,0,0]]):
    if n==0: return(1)
    mv1 = [plus(P[-1],x) for x in mo]
    mv2=[x for x in mv1 if x not in P]
    if n==1: return(len(mv2))
    else: return(sum(a(n-1,add(P,x)) for x in mv2))
[a(n) for n in range(8)]
# Robert FERREOL, Nov 30 2018

A078717 Number of n-step self-avoiding walks on cubic lattice with first step specified.

Original entry on oeis.org

1, 5, 25, 121, 589, 2821, 13565, 64661, 308981, 1468313, 6989025, 33140457, 157329085, 744818613, 3529191009, 16686979329, 78955042017, 372953947349, 1762672203269, 8319554639789, 39285015083693, 185296997240401, 874331369198569

Offset: 1

Hugo Pfoertner, Dec 18 2002

nonn

See A001412 for references.

Hugo Pfoertner, Table of n, a(n) for n = 1..36

Equals A001412/6. Cf. A001411, A046661, A002902.

A140476 Number of self-avoiding walks on cubic lattice with no more than n steps.

Original entry on oeis.org

1, 7, 37, 187, 913, 4447, 21373, 102763, 490729, 2344615, 11154493, 53088643, 251931385, 1195905895, 5664817573, 26839963627, 126961839601, 600692091703, 2838415775797, 13414448995411, 63331776834145, 299041867336303

Offset: 0

Jonathan Vos Post, Jun 29 2008

nonn

Primes include a(1) = 7, a(2) = 37, a(5) = 4447, a(8) = 102763, a(15) = 26839963627.

			a(9) = 1 + 6 + 30 + 150 + 726 + 3534 + 16926 + 81390 + 387966 + 1853886 = 2344615.

Partial sums of A001412.

Cf. A001411, A001412, A001413, A002902, A078717.

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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A001412 Number of n-step self-avoiding walks on cubic lattice.

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A078717 Number of n-step self-avoiding walks on cubic lattice with first step specified.

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A140476 Number of self-avoiding walks on cubic lattice with no more than n steps.

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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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A213453 Self-avoiding walks on the f.c.c. lattice.

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