A192208 -id:A192208 - cp's OEIS frontend

A001334 Number of n-step self-avoiding walks on hexagonal [ =triangular ] lattice.

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

N. J. A. Sloane

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

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.

Cf. A001411, A001412, A001336, A001666, A001335, A036418, A192208.

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

A192871 Number of n-step prudent self-avoiding walks on honeycomb lattice.

Original entry on oeis.org

1, 3, 6, 12, 24, 48, 90, 168, 318, 594, 1092, 2004, 3678, 6720, 12210, 22128, 40074, 72372, 130380, 234432, 421128, 755208, 1352328, 2418246, 4320552, 7709898, 13744764, 24477618, 43560444, 77448330, 137602440, 244277016, 433399824, 768379830, 1361530134

Offset: 0

Alois P. Heinz, Jul 11 2011

A prudent walk never takes a step pointing towards a vertex it has already visited. Prudent walks are self-avoiding but not reversible in general.

			This 8-step prudent self-avoiding walk on honeycomb lattice from (S) to (E) is not reversible:
.           o...o       o...o
.          .     .     .     .
.     o...o       4---3       o
.    .     .     /     \     .
.   o       6---5       2...o
.    .     /     .     /     .
.     o...7      (S)--1       o
.    .     \     .     .     .
.   o      (E)..o       o...o
.    .     .     .     .
.     o...o       o...0

Alois P. Heinz, Table of n, a(n) for n = 0..110
Mireille Bousquet-Mélou, Families of prudent self-avoiding walks, DMTCS proc. AJ, 2008, 167-180.
Mireille Bousquet-Mélou, Families of prudent self-avoiding walks, arXiv:0804.4843 [math.CO], 2008-2009.
Enrica Duchi, On some classes of prudent walks, in: FPSAC'05, Taormina, Italy, 2005.

Cf. A001668, A006851, A192208.

i:= n-> max(n, 0)+1: d:= n-> max(n-1, -1):
b:= proc(n, x, y, z, u, v, w) option remember;
    `if`(n=0, 1, `if`(x>y, b(n, y, x, w, v, u, z),
    `if`(min(y, z)<=0 or x=-1,
        b(n-1, d(y), d(z), u, i(v), i(w), x), 0)+
    `if`(min(w, x)<=0 or y=-1,
        b(n-1, d(w), d(x), y, i(z), i(u), v), 0)))
    end:
a:= n-> `if`(n<2, 1 +2*n, 6*b(n-2, -1, -1, 1, 2, 1, -1)):
seq(a(n), n=0..20);

i[n_] := Max[n, 0] + 1; d[n_] := Max[n - 1, -1];
b[n_, x_, y_, z_, u_, v_, w_] := b[n, x, y, z, u, v, w] = If[n == 0, 1, If[x > y, b[n, y, x, w, v, u, z], If[Min[y, z] <= 0 || x == -1, b[n - 1, d[y], d[z], u, i[v], i[w], x], 0] + If[Min[w, x] <= 0 || y == -1, b[n - 1, d[w], d[x], y, i[z], i[u], v], 0]]];
a[n_] := If[n < 2, 1 + 2 n, 6 b[n - 2, -1, -1, 1, 2, 1, -1]];
a /@ Range[0, 34] (* Jean-François Alcover, Sep 22 2019, after Alois P. Heinz *)

cp's OEIS Frontend

A001334 Number of n-step self-avoiding walks on hexagonal [ =triangular ] lattice.

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