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A002931
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Number of self-avoiding polygons of length 2n on square lattice (not allowing rotations).
(Formerly M1780 N0703)
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33
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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
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OFFSET
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1,3
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COMMENTS
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Translations are allowed, but not rotations or 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
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REFERENCES
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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)]
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.
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.
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)]
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)]
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).
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LINKS
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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)
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EXAMPLE
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At length 8 there are 7 polygons, consisting of the 2, 1, 4 resp. rotations of:
._. .___. .___.
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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
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CROSSREFS
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Cf. A302335 (constant coefficients in p(k,n)).
Cf. A302336 (linear coefficients in p(k,n)).
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KEYWORD
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nonn,walk,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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