A002931 - OEIS

%I M1780 N0703 #69 Mar 18 2021 23:08:21

%S 0,1,2,7,28,124,588,2938,15268,81826,449572,2521270,14385376,83290424,

%T 488384528,2895432660,17332874364,104653427012,636737003384,

%U 3900770002646,24045500114388,149059814328236,928782423033008,5814401613289290,36556766640745936

%N Number of self-avoiding polygons of length 2n on square lattice (not allowing rotations).

%C Translations are allowed, but not rotations or reflections.

%C a(n) is also the coefficient of n^2 in the sequence of quadratic polynomials giving the numbers of 2k-cycles in the n X n grid graph for n >= k-1 (see the example). - _Eric W. Weisstein_, Apr 05 2018

%D N. Clisby and I. Jensen: A new transfer-matrix algorithm for exact enumerations: self-avoiding polygons on the square lattice, J. Phys. A: Math. Theor. 45 (2012). Also arXiv:1111.5877, 2011. [Extends sequence to a(65)]

%D I. G. Enting: Generating functions for enumerating self-avoiding rings on the square lattice, J. Phys. A: Math. Gen. 13 (1980).  pp. 3713-3722. See Table 2.

%D A. J. Guttmann, Asymptotic analysis of power-series expansions, pp. 1-234 of C. Domb and J. L. Lebowitz, editors, Phase Transitions and Critical Phenomena. Vol. 13, Academic Press, NY, 1989.

%D B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 461.

%D I. Jensen: A parallel algorithm for the enumeration of self-avoiding polygons on the square lattice, J. Phys. A: Math. Gen. 36 (2003). [Extends sequence to a(55)]

%D I. Jensen and A. J. Guttmann: Self-avoiding polygons on the square lattice, J. Phys. A: Math. Gen. 32 (1999). Also arXiv:cond-mat/9905291. [Extends sequence to a(45)]

%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 N. J. A. Sloane, <a href="/A002931/b002931.txt">Table of n, a(n) for n = 1..65</a> [Formed from tables in several references, the most recent being Clisby-Jensen, 2011/2012]

%H Jérôme Bastien, <a href="http://arxiv.org/abs/1603.08775">Construction and enumeration of circuits capable of guiding a miniature vehicle</a>, arXiv:1603.08775 [math.CO], 2016. Cites this sequence.

%H Nathan Clisby, <a href="https://clisby.net/entingfest/talks/clisby_entingfest.pdf">Lattice enumeration</a>, Slides of talk at Enting fest, CSIRO, Aspendale, 2015; <a href="/A002931/a002931.pdf">Lattice enumeration</a> [Local copy].

%H M. E. Fisher and D. S. Gaunt, <a href="http://dx.doi.org/10.1103/PhysRev.133.A224">Ising model and self-avoiding walks on hypercubical lattices and high density expansions</a>, Phys. Rev. 133 (1964) A224-A239.

%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 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 364.

%H A. J. Guttmann, <a href="/A002906/a002906.pdf">Asymptotic analysis of power-series expansions</a>, pp. 1-13, 56-57, 142-143, 150-151 from of C. Domb and J. L. Lebowitz, editors, Phase Transitions and Critical Phenomena. Vol. 13, Academic Press, NY, 1989. (Annotated scanned copy)

%H A. J. Guttmann and I. G. Enting, <a href="https://doi.org/10.1088/0305-4470/21/3/009">The size and number of rings on the square lattice</a>, J. Phys. A 21 (1988), L165-L172.

%H Brian Hayes, <a href="https://www.jstor.org/stable/27857052">How to avoid yourself</a>, American Scientist 86 (1998) 314-319.

%H B. J. Hiley and M. F. Sykes, <a href="http://dx.doi.org/10.1063/1.1701041">Probability of initial ring closure in the restricted random-walk model of a macromolecule</a>, J. Chem. Phys., 34 (1961), 1531-1537.

%H I. Jensen, <a href="https://doi.org/10.1088/0305-4470/36/21/304">A parallel algorithm for the enumeration of self-avoiding polygons on the square lattice</a>, Journal of Physics A, Vol. 36 (2003), pp. 5731-5745.

%H I. Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/polygons/series/square.perim.ser">More terms</a>

%H G. S. Rushbrooke and J. Eve, <a href="http://dx.doi.org/10.1063/1.1730595">On Noncrossing Lattice Polygons</a>, Journal of Chemical Physics, 31 (1959), 1333-1334.

%H S. G. Whittington and J. P. Valleau, <a href="https://doi.org/10.1088/0305-4470/3/1/003">Figure eights on the square lattice: enumeration and Monte Carlo estimation</a>, J. Phys. A 3 (1970), 21-27.  See Table 2.

%e At length 8 there are 7 polygons, consisting of the 2, 1, 4 resp. rotations of:

%e ._. .___. .___.

%e | | | . | | ._|

%e | | |___| |_|

%e |_|

%e Let p(k,n) be the number of 2k-cycles in the n X n grid graph for n >= k-1.  p(k,n) are quadratic polynomials in n, with the first few given by:

%e p(1,n) = 0,

%e p(2,n) = 1 - 2*n + n^2,

%e p(3,n) = 4 - 6*n + 2*n^2,

%e p(4,n) = 26 - 28*n + 7*n^2,

%e p(5,n) = 164 - 140*n + 28*n^2,

%e p(6,n) = 1046 - 740*n + 124*n^2,

%e p(7,n) = 6672 - 4056*n + 588*n^2,

%e p(8,n) = 42790 - 22904*n + 2938*n^2,

%e p(9,n) = 275888 - 132344*n + 15268*n^2,

%e ...

%e The quadratic coefficients give a(n), so the first few are 0, 1, 2, 7, 28, 124, .... - _Eric W. Weisstein_, Apr 05 2018

%Y Cf. A056634, A036638, A036639. Equals A010566(n)/(4n).

%Y Cf. A057730.

%Y Cf. A302335 (constant coefficients in p(k,n)).

%Y Cf. A302336 (linear coefficients in p(k,n)).

%K nonn,walk,nice

%O 1,3

%A _N. J. A. Sloane_

%E Updated by _N. J. A. Sloane_, Mar 18 2021