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

Author

N. J. A. Sloane

Keywords

• nonn
• walk
• nice

References

• 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

• 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]

Crossrefs

Cf. A006851.

Programs

Maple

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

Mathematica

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 *)

Formula

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

Extensions

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