cp's OEIS Frontend

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

%I A001668 M2559 N1013 #60 Aug 08 2024 11:06:25
%S A001668 1,3,6,12,24,48,90,174,336,648,1218,2328,4416,8388,15780,29892,56268,
%T A001668 106200,199350,375504,704304,1323996,2479692,4654464,8710212,16328220,
%U A001668 30526374,57161568,106794084,199788408,372996450,697217994,1300954248
%N A001668 Number of self-avoiding n-step walks on honeycomb lattice.
%D A001668 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001668 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001668 Hugo Pfoertner, <a href="/A001668/b001668.txt">Table of n, a(n) for n = 0..105</a>
%H A001668 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 A001668 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 A001668 E. J. Janse van Rensburg and S. G. Whittington, <a href="https://arxiv.org/abs/2407.21112">Exponential growth rate of lattice comb polymers</a>, arXiv:2407.21112 [cond-mat.stat-mech], 2024. See p. 9.
%H A001668 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 A001668 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 A001668 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 A001668 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 A001668 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 A001668 a:= proc(n) local v, b;
%p A001668       if n<2 then return 1 +2*n fi;
%p A001668       v:= proc() false end: v(0, 0), v(1, 0):= true$2;
%p A001668       b:= proc(n, x, y) local c;
%p A001668             if v(x, y) then 0
%p A001668           elif n=0 then 1
%p A001668           else v(x, y):= true;
%p A001668                c:= b(n-1, x+1, y) + b(n-1, x-1, y) +
%p A001668                    b(n-1, x, y-1+2*((x+y) mod 2));
%p A001668                v(x, y):= false; c
%p A001668             fi
%p A001668           end;
%p A001668       6*b(n-2, 1, 1)
%p A001668     end:
%p A001668 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jul 07 2011
%t A001668 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 *)
%Y A001668 Cf. A006851.
%K A001668 nonn,walk,nice
%O A001668 0,2
%A A001668 _N. J. A. Sloane_
%E A001668 More terms from Pab Ter (pabrlos(AT)yahoo.com), May 06 2004