A001334
Number of n-step self-avoiding walks on hexagonal [ =triangular ] lattice.
Original entry on oeis.org
1, 6, 30, 138, 618, 2730, 11946, 51882, 224130, 964134, 4133166, 17668938, 75355206, 320734686, 1362791250, 5781765582, 24497330322, 103673967882, 438296739594, 1851231376374, 7812439620678, 32944292555934, 138825972053046
Offset: 0
- A. J. Guttmann, Asymptotic analysis of power-series expansions, pp. 1-234 of C. Domb and J. L. Lebowitz, editors, Phase Transitions and Critical Phenomena. Vol. 13, Academic Press, NY, 1989.
- B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 459.
- 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).
- I. Jensen, Table of n, a(n) for n = 0..40 (from the Jensen link below)
- Sergio Caracciolo, Anthony J. Guttmann, Iwan Jensen, Andrea Pelissetto, Andrew N. Rogers, Alan D. Sokal, Correction-to-Scaling Exponents for Two-Dimensional Self-Avoiding Walks, Journal of Statistical Physics, September 2005, Volume 120, Issue 5, pp. 1037-1100.
- 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, Asymptotic analysis of power-series expansions, pp. 1-13, 56-57, 142-143, 150-151 from of C. Domb and J. L. Lebowitz, editors, Phase Transitions and Critical Phenomena. Vol. 13, Academic Press, NY, 1989. (Annotated scanned copy)
- A. J. Guttmann and J. Wang, The extension of self-avoiding random walk series in two dimensions, J. Phys. A 24 (1991), 3107-3109.
- B. D. Hughes, Random Walks and Random Environments, vol. 1, Oxford 1995, Tables and references for self-avoiding walks counts [Annotated scanned copy of several pages of a preprint or a draft of chapter 7 "The self-avoiding walk"]
- I. Jensen, Series Expansions for Self-Avoiding Walks
- J. L. Martin, M. F. Sykes and F. T. Hioe, Probability of initial ring closure for self-avoiding walks on the face-centered cubic and triangular lattices, J. Chem. Phys., 46 (1967), 3478-3481.
- G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
- D. C. Rapaport, End-to-end distance of linear polymers in two dimensions: a reassessment, J. Phys. A 18 (1985), L201.
- S. Redner, Distribution functions in the interior of polymer chains, J. Phys. A 13 (1980), 3525-3541, doi:10.1088/0305-4470/13/11/023.
- 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.
- Joris van der Hoeven, On asymptotic extrapolation, Journal of symbolic computation, 2009, p. 1010.
-
mo={{2, 0},{-1, 1},{-1, -1},{-2, 0},{1, -1},{1, 1}}; a[0]=1;
a[tg_, p_:{{0, 0}}] := Block[{e, mv = Complement[Last[p]+# & /@ mo, p]}, If[tg == 1, Length@mv, Sum[a[tg-1, Append[p, e]], {e, mv}]]];
a /@ Range[0, 6]
(* Robert FERREOL, Nov 28 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=[[2,0],[-1,1],[-1, -1],[-2,0],[1,-1],[1, 1]]
def a(n,P=[[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(11)]
# Robert FERREOL, Dec 11 2018
A001336
Number of n-step self-avoiding walks on f.c.c. lattice.
Original entry on oeis.org
1, 12, 132, 1404, 14700, 152532, 1573716, 16172148, 165697044, 1693773924, 17281929564, 176064704412, 1791455071068, 18208650297396, 184907370618612, 1876240018679868, 19024942249966812, 192794447005403916, 1952681556794601732, 19767824914170222996
Offset: 0
- B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 460.
- 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..24 (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.
- B. D. Hughes, Random Walks and Random Environments, vol. 1, Oxford 1995, Tables and references for self-avoiding walks counts [Annotated scanned copy of several pages of a preprint or a draft of chapter 7 "The self-avoiding walk"]
- J. L. Martin, M. F. Sykes and F. T. Hioe, Probability of initial ring closure for self-avoiding walks on the face-centered cubic and triangular lattices, J. Chem. Phys., 46 (1967), 3478-3481.
- S. McKenzie, Self-avoiding walks on the face-centered cubic lattice, J. Phys. A 12 (1979), L267-L270.
- S. Redner, Distribution functions in the interior of polymer chains, J. Phys. A 13 (1980), 3525-3541, doi:10.1088/0305-4470/13/11/023.
- 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 II.
- 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.
- Index entries for sequences related to f.c.c. lattice
A336906
Number of n-step self-avoiding walks on the b.c.c. lattice with no non-contiguous adjacencies.
Original entry on oeis.org
1, 8, 56, 296, 1640, 8984, 49256, 266600, 1448072, 7820984, 42316952, 227940584, 1229803016, 6612947048, 35605181720, 191204813288, 1027868658200
Offset: 0
A002903
Number of n-step self-avoiding walks on b.c.c. lattice (version 1).
Original entry on oeis.org
1, 4, 28, 196, 1324, 8980, 60028, 402412, 2675860, 17826340, 118145548, 784024780, 5184334996, 34313323804, 226516271020, 1496391824212, 9865667928796, 65080520041804, 428641139406628, 2824446024265444, 18587519784608836, 122369125319060884
Offset: 0
- 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).
Equals
A001666/2 apart from initial term.
A001667
2n-step polygons on b.c.c. lattice.
Original entry on oeis.org
96, 1776, 43776, 1237920, 37903776, 1223681760, 41040797376, 1416762272736, 50027402384640, 1799035070369856
Offset: 2
- 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).
- P. Butera and M. Comi, Enumeration of the self-avoiding polygons on a lattice by the Schwinger-Dyson equations, Annals of Combinatorics 3, 277-286 (1999); arXiv:cond-mat/9903297, 1999.
- M. E. Fisher and M. F. Sykes, Excluded-volume problem and the Ising model of ferromagnetism, Phys. Rev. 114 (1959), 45-58.
- M. F. Sykes et al., The number of self-avoiding walks on a lattice, J. Phys. A 5 (1972), 661-666.
- Index entries for sequences related to b.c.c. lattice
A260347
Coefficients of the sum of the mean squared distance generating function for the body-centered cubic lattice.
Original entry on oeis.org
3, 48, 531, 5088, 44751, 373404, 2999985, 23457672, 179561859, 1352017596, 10042445889, 73771019064, 536817918837, 3875387231484, 27783517769223, 197998094612568
Offset: 1
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