A006851 -id:A006851 - cp's OEIS frontend

A001668 Number of self-avoiding n-step walks on honeycomb lattice.

1, 3, 6, 12, 24, 48, 90, 174, 336, 648, 1218, 2328, 4416, 8388, 15780, 29892, 56268, 106200, 199350, 375504, 704304, 1323996, 2479692, 4654464, 8710212, 16328220, 30526374, 57161568, 106794084, 199788408, 372996450, 697217994, 1300954248

Offset: 0

N. J. A. Sloane

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).

Hugo Pfoertner, Table of n, a(n) for n = 0..105
H. Duminil-Copin and S. Smirnov, The connective constant of the honeycomb lattice equals sqrt(2+sqrt(2)), Ann. Math. 175 (2012), pp. 1653-1665. arXiv:1007.0575 [math-ph], 2010-2011.
M. E. Fisher and M. F. Sykes, Excluded-volume problem and the Ising model of ferromagnetism, Phys. Rev. 114 (1959), 45-58.
E. J. Janse van Rensburg and S. G. Whittington, Exponential growth rate of lattice comb polymers, arXiv:2407.21112 [cond-mat.stat-mech], 2024. See p. 9.
I. Jensen, Series Expansions for Self-Avoiding Walks [Gives 105 terms] - _Hugo Pfoertner_, Aug 10 2014
D. MacDonald, D. L. Hunter, K. Kelly and N. Jan, Self-avoiding walks in two to five dimensions: exact enumerations and series study, J Phys A: Math Gen 25 (1992) 1429-1440. [Gives 42 terms]
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 et al., The asymptotic behavior of selfavoiding walks and returns on a lattice, J. Phys. A 5 (1972), 653-660. [Gives 34 terms]

Cf. A006851.

a:= proc(n) local v, b;
      if n<2 then return 1 +2*n fi;
      v:= proc() false end: v(0, 0), v(1, 0):= true$2;
      b:= proc(n, x, y) local c;
            if v(x, y) then 0
          elif n=0 then 1
          else v(x, y):= true;
               c:= b(n-1, x+1, y) + b(n-1, x-1, y) +
                   b(n-1, x, y-1+2*((x+y) mod 2));
               v(x, y):= false; c
            fi
          end;
      6*b(n-2, 1, 1)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 07 2011

a[n_] := a[n] = Module[{v, b}, If[n < 2 , Return[1+2*n]]; v[0, 0] = v[1, 0] = True; v[, ] = False; b[m_, x_, y_] := Module[{c}, If[v[x, y], 0 , If[ m == 0 , 1, v[x, y] = True; c = b[m-1, x+1, y] + b[m-1, x-1, y] + b[m-1, x, y-1 + 2*Mod[x+y, 2]]; v[x, y] = False; c]]]; 6*b[n-2, 1, 1]]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 32}] (* Jean-François Alcover, Nov 25 2013, translated from Alois P. Heinz's Maple program *)

mu^n <= a(n) <= mu^n alpha^sqrt(n) for mu = A179260 and some alpha. It has been conjectured that a(n) ~ mu^n * n^(11/32). - Charles R Greathouse IV, Nov 08 2013

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 06 2004

A006817 Trails of length n on square lattice.

Original entry on oeis.org

1, 4, 12, 36, 108, 316, 916, 2628, 7500, 21268, 60092, 169092, 474924, 1329188, 3715244, 10359636, 28856252, 80220244, 222847804, 618083972, 1713283628, 4742946484, 13123882524, 36274940740, 100226653420, 276669062116, 763482430316, 2105208491748, 5803285527724

Offset: 0

N. J. A. Sloane

A trail is a path which may cross itself but does not reuse an edge. This sequence counts directed paths on the square lattice.

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..31 (from Conway & Guttmann; terms 0..30 from Bert Dobbelaere)
A. R. Conway and A. J. Guttmann, Enumeration of self-avoiding trails on a square lattice using a transfer matrix technique, J. Phys. A: Math. Gen., 26 (1993), 1535-1552; arXiv:hep-lat/9211063, 1992.
A. J. Guttmann, Lattice trails II: numerical results, J. Phys. A 22 (1989), 575-588.
A. Malakis, The trail problem on the square lattice, J. Phys A 9 (8) (1976) p 1283. Table 1.

Undirected trails-rotation and reflection are counted by A001997.

Cf. A006818, A006819, A006851.

More terms from David W. Wilson, Jul 20 2001

More terms from Bert Dobbelaere, Jan 19 2019

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 *)

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A001668 Number of self-avoiding n-step walks on honeycomb lattice.

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A006817 Trails of length n on square lattice.

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

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