A001668 Number of self-avoiding n-step walks on honeycomb lattice. (Formerly M2559 N1013)

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

0,2

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.

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]

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

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 *)

Crossrefs

Cf. A006851.

Sequence in context: A102255 A192871 A002910 * A080616 A090572 A163876

Adjacent sequences: A001665 A001666 A001667 * A001669 A001670 A001671

Keyword

nonn,walk,nice

Author

N. J. A. Sloane

Extensions

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

Status

approved