login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A002931 Number of self-avoiding polygons of length 2n on square lattice (not allowing rotations).
(Formerly M1780 N0703)
21
0, 1, 2, 7, 28, 124, 588, 2938, 15268, 81826, 449572, 2521270, 14385376, 83290424, 488384528, 2895432660, 17332874364, 104653427012, 636737003384, 3900770002646, 24045500114388, 149059814328236, 928782423033008, 5814401613289290, 36556766640745936 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Translations are allowed, but not rotations and reflections.

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

REFERENCES

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.

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

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

I. Jensen, Table of n, a(n) for n = 1..55 (from link below)

Jérôme Bastien, Construction and enumeration of circuits capable of guiding a miniature vehicle, arXiv:1603.08775 [math.CO], 2016. Cites this sequence.

M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev. 133 (1964) A224-A239.

M. E. Fisher and M. F. Sykes, Excluded-volume problem and the Ising model of ferromagnetism, Phys. Rev. 114 (1959), 45-58.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 364.

A. J. Guttmann, Asymptotic analysis of power-series expansions, 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)

A. J. Guttmann and I. G. Enting, The size and number of rings on the square lattice, J. Phys. A 21 (1988), L165-L172.

Brian Hayes, How to avoid yourself, American Scientist 86 (1998) 314-319.

B. J. Hiley and M. F. Sykes, Probability of initial ring closure in the restricted random-walk model of a macromolecule, J. Chem. Phys., 34 (1961), 1531-1537.

I. Jensen, A parallel algorithm for the enumeration of self-avoiding polygons on the square lattice, Journal of Physics A, Vol. 36 (2003), pp. 5731-5745.

I. Jensen, More terms

G. S. Rushbrooke and J. Eve, On Noncrossing Lattice Polygons, Journal of Chemical Physics, 31 (1959), 1333-1334.

S. G. Whittington and J. P. Valleau, Figure eights on the square lattice: enumeration and Monte Carlo estimation, J. Phys. A 3 (1970), 21-27.  See Table 2.

EXAMPLE

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

._. .___. .___.

| | | . | | ._|

| | |___| |_|

|_|

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:

p(1,n) = 0,

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

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

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

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

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

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

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

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

...

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

CROSSREFS

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

Cf. A057730.

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

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

Sequence in context: A143927 A253787 A060379 * A088702 A112565 A227845

Adjacent sequences:  A002928 A002929 A002930 * A002932 A002933 A002934

KEYWORD

nonn,walk,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

Has been extended to 55 terms by Jensen (2003). - Markus Voege (markus.voege(AT)inria.fr), Nov 23 2003

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 20 12:38 EST 2019. Contains 320327 sequences. (Running on oeis4.)