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

Author

N. J. A. Sloane

Keywords

• nonn
• walk
• nice

References

• 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).

Links

• 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.

Crossrefs

Cf. A002902, A078717, A001411, A001413.

Programs

Mathematica

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 *)

Python

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