cp's OEIS Frontend

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

%I A001412 M4202 N1754 #78 Aug 05 2024 13:39:08
%S A001412 1,6,30,150,726,3534,16926,81390,387966,1853886,8809878,41934150,
%T A001412 198842742,943974510,4468911678,21175146054,100121875974,473730252102,
%U A001412 2237723684094,10576033219614,49917327838734,235710090502158,1111781983442406,5245988215191414,24730180885580790,116618841700433358,549493796867100942,2589874864863200574,12198184788179866902,57466913094951837030,270569905525454674614
%N A001412 Number of n-step self-avoiding walks on cubic lattice.
%D A001412 Steven R. Finch, Mathematical Constants, Cambridge, 2003, section 5.10, pp. 331-339.
%D A001412 B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 462.
%D A001412 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001412 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001412 R. D. Schram, G. T. Barkema, and R. H. Bisseling, <a href="/A001412/b001412.txt">Table of n, a(n) for n = 0..36</a>
%H A001412 N. Clisby, <a href="https://www.ams.org/journals/notices/202104/rnoti-p504.pdf">Enumerative combinatorics of lattice polymers</a>, Notices AMS, 68:4 (2021), 504-515. (Excellent survey)
%H A001412 N. Clisby, R. Liang, and G. Slade, <a href="https://doi.org/10.1088/1751-8113/40/36/003">Self-avoiding walk enumeration via the lace expansion</a>, J. Phys. A: Math. Theor. 40 (2007), 10973-11017, Table A5 for n<=30.
%H A001412 Steven R. Finch, <a href="http://web.archive.org/web/20010207195347/http://www.mathsoft.com/asolve/constant/cnntv/cnntv.html">Self-Avoiding-Walk Connective Constants</a>
%H A001412 M. E. Fisher and M. F. Sykes, <a href="https://doi.org/10.1103/PhysRev.114.45">Excluded-volume problem and the Ising model of ferromagnetism</a>, Phys. Rev. 114 (1959), 45-58.
%H A001412 A. J. Guttmann, <a href="https://doi.org/10.1088/0305-4470/20/7/029">On the critical behavior of self-avoiding walks</a>, J. Phys. A 20 (1987), 1839-1854.
%H A001412 B. J. Hiley and M. F. Sykes, <a href="https://doi.org/10.1063/1.1701041">Probability of initial ring closure in the restricted random-walk model of a macromolecule</a>, J. Chem. Phys., 34 (1961), 1531-1537.
%H A001412 D. S. McKenzie and C. Domb, <a href="https://doi.org/10.1088/0370-1328/92/3/316">The second osmotic virial coefficient of athermal polymer solutions</a>, Proceedings of the Physical Society, 92 (1967) 632-649.
%H A001412 A. M. Nemirovsky, Karl F. Freed, Takao Ishinabe, and Jack F. Douglas, <a href="https://doi.org/10.1007/BF01049010">Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers</a>, J. Statist. Phys., 67 (1992), 1083-1108.
%H A001412 D. Randall, <a href="https://citeseerx.ist.psu.edu/pdf/bfc5b51d2ee76c6a7d347b2ac42ef16f040fbeb8">Counting in Lattices: Combinatorial Problems from Statistical Mechanics</a>, PhD Thesis (1994).
%H A001412 Raoul D. Schram, Gerard T. Barkema, and Rob H. Bisseling, <a href="https://arxiv.org/abs/1104.2184">Exact enumeration of self-avoiding walks</a>, arXiv:1104.2184 [math-ph], 2011.
%H A001412 Nobu C. Shirai and Naoyuki Sakumichi, <a href="https://arxiv.org/abs/2202.12483">Negative Energetic Elasticity of Lattice Polymer Chain in Solvent</a>, arXiv:2202.12483 [cond-mat.soft], 2022.
%H A001412 M. F. Sykes, <a href="https://doi.org/10.1063/1.1724212">Some counting theorems in the theory of the Ising problem and the excluded volume problem</a>, J. Math. Phys., 2 (1961), 52-62.
%H A001412 M. F. Sykes, <a href="https://doi.org/10.1063/1.1734262">Self-avoiding walks on the simple cubic lattice</a>, J. Chem. Phys., 39 (1963), 410-411.
%H A001412 M. F. Sykes, A. J. Guttmann, M. G. Watts, and P. D. Roberts, <a href="https://doi.org/10.1088/0305-4470/5/5/006">The asymptotic behavior of self-avoiding walks and returns on a lattice</a>, J. Phys. A 5 (1972), 653-660.
%H A001412 M. F. Sykes, D. S. McKenzie, M. G. Watts, and J. L. Martin, <a href="http://dx.doi.org/10.1088/0305-4470/5/5/007">The number of self-avoiding rings on a lattice</a>, J. Phys. A 5 (1972), 661-666.
%t A001412 mo = {{1, 0, 0}, {-1, 0, 0}, {0, 1, 0}, {0, -1, 0}, {0, 0, 1}, {0, 0, -1}}; a[0] = 1;
%t A001412 a[tg_, p_: {{0, 0, 0}}] := Block[{e, mv = Complement[Last[p] + # & /@ mo, p]},
%t A001412 If[tg == 1, Return[Length@mv],Sum[a[tg - 1, Append[p, e]], {e, mv}]]];
%t A001412 a /@ Range[0, 8]
%t A001412 (* _Robert FERREOL_, Nov 30 2018, after the program of _Giovanni Resta_ in A001411 *)
%o A001412 (Python)
%o A001412 def add(L,x):
%o A001412     M=[y for y in L];M.append(x)
%o A001412     return(M)
%o A001412 plus=lambda L,M : [x+y for x,y in zip(L,M)]
%o A001412 mo=[[1,0,0],[-1,0,0],[0,1,0],[0,-1,0],[0,0,1],[0,0,-1]]
%o A001412 def a(n,P=[[0,0,0]]):
%o A001412     if n==0: return(1)
%o A001412     mv1 = [plus(P[-1],x) for x in mo]
%o A001412     mv2=[x for x in mv1 if x not in P]
%o A001412     if n==1: return(len(mv2))
%o A001412     else: return(sum(a(n-1,add(P,x)) for x in mv2))
%o A001412 [a(n) for n in range(8)]
%o A001412 # _Robert FERREOL_, Nov 30 2018
%Y A001412 Cf. A002902, A078717, A001411, A001413.
%K A001412 nonn,walk,nice
%O A001412 0,2
%A A001412 _N. J. A. Sloane_