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

%I M2559 N1013

%S 1,3,6,12,24,48,90,174,336,648,1218,2328,4416,8388,15780,29892,56268,

%T 106200,199350,375504,704304,1323996,2479692,4654464,8710212,16328220,

%U 30526374,57161568,106794084,199788408,372996450,697217994,1300954248

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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Hugo Pfoertner, <a href="/A001668/b001668.txt">Table of n, a(n) for n = 0..105</a>

%H H. Duminil-Copin and S. Smirnov, <a href="http://arxiv.org/abs/1007.0575">The connective constant of the honeycomb lattice equals sqrt(2+sqrt(2))</a>, Ann. Math. 175 (2012), pp. 1653-1665. arXiv:1007.0575 [math-ph], 2010-2011.

%H M. E. Fisher and M. F. Sykes, <a href="http://dx.doi.org/10.1103/PhysRev.114.45">Excluded-volume problem and the Ising model of ferromagnetism</a>, Phys. Rev. 114 (1959), 45-58.

%H I. Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/saw/SAW_ser.html">Series Expansions for Self-Avoiding Walks</a> [Gives 105 terms] - _Hugo Pfoertner_, Aug 10 2014

%H D. MacDonald, D. L. Hunter, K. Kelly and N. Jan, <a href="http://dx.doi.org/10.1088/0305-4470/25/6/006">Self-avoiding walks in two to five dimensions: exact enumerations and series study</a>, J Phys A: Math Gen 25 (1992) 1429-1440. [Gives 42 terms]

%H M. F. Sykes, <a href="http://dx.doi.org/10.1063/1.1724212">Some counting theorems in the theory of the Ising problem and the excluded volume problem</a>, J. Math. Phys., 2 (1961), 52-62.

%H M. F. Sykes et al., <a href="http://dx.doi.org/10.1088/0305-4470/5/5/007">The asymptotic behavior of selfavoiding walks and returns on a lattice</a>, J. Phys. A 5 (1972), 653-660. [Gives 34 terms]

%F 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

%p a:= proc(n) local v, b;

%p if n<2 then return 1 +2*n fi;

%p v:= proc() false end: v(0, 0), v(1, 0):= true$2;

%p b:= proc(n, x, y) local c;

%p if v(x, y) then 0

%p elif n=0 then 1

%p else v(x, y):= true;

%p c:= b(n-1, x+1, y) + b(n-1, x-1, y) +

%p b(n-1, x, y-1+2*((x+y) mod 2));

%p v(x, y):= false; c

%p fi

%p end;

%p 6*b(n-2, 1, 1)

%p end:

%p seq(a(n), n=0..20); # _Alois P. Heinz_, Jul 07 2011

%t 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-Fran├žois Alcover_, Nov 25 2013, translated from _Alois P. Heinz_'s Maple program *)

%Y Cf. A006851.

%K nonn,walk,nice

%O 0,2

%A _N. J. A. Sloane_

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

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Last modified June 22 17:18 EDT 2021. Contains 345388 sequences. (Running on oeis4.)