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A001411 Number of n-step self-avoiding walks on square lattice.
(Formerly M3448 N1402)
22
1, 4, 12, 36, 100, 284, 780, 2172, 5916, 16268, 44100, 120292, 324932, 881500, 2374444, 6416596, 17245332, 46466676, 124658732, 335116620, 897697164, 2408806028, 6444560484, 17266613812, 46146397316, 123481354908, 329712786220, 881317491628 (list; graph; refs; listen; history; internal format)
OFFSET

0,2

COMMENTS

a(0)+1 = 2 is prime, a(1)+1 = 5 is prime, a(2)+1 = 13 is prime, a(3)+1 = 37 is prime, a(4)+1 = 101 is prime, a(10)+1 = 44101 is prime, a(11)+1 = 120293 is prime, a(27)+1 = 881317491629 is prime. - Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 16 2005

REFERENCES

B. Bollobas and O. Riordan, Percolation, Cambridge, 2006, see p. 15.

A. R. Conway et al., Algebraic techniques for enumerating self-avoiding walks ..., J. Phys A 26 (1993) 1519-1534.

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 331-339.

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

A. J. Guttmann, On the critical behavior of self-avoiding walks, J. Phys. A 20 (1987), 1839-1854.

A. J. Guttmann and A. R. Conway, Self-Avoiding Walks and Polygons, Annals of Combinatorics 5 (2001) 319-345.

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.

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

G. Slade, Self-avoiding walks, Math. Intelligencer, 16 (No. 1, 1994), 29-35.

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).

M. F. Sykes, Some counting theorems in the theory of the Ising problem and the excluded volume problem, J. Math. Phys., 2 (1961), 52-62.

M. F. Sykes et al., The asymptotic behavior of selfavoiding walks and returns on a lattice, J. Phys. A 5 (1972), 653-660.

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..71 [from the Jensen link below]

H. Bottomley, Illustration of initial terms

S. R. Finch, Self-Avoiding-Walk Connective Constants

I. Jensen, Series Expansions for Self-Avoiding Walks

D. Randall, Counting in Lattices: Combinatorial Problems from Statistical Mechanics, PhD Thesis.

CROSSREFS

Twice A002900.

Sequence in context: A192626 A002906 A191756 * A095350 A084776 A162921

Adjacent sequences:  A001408 A001409 A001410 * A001412 A001413 A001414

KEYWORD

nonn,walk,nice

AUTHOR

N. J. A. Sloane (njas(AT)research.att.com), A. J. Guttmann (tonyg(AT)maths.mu.OZ.AU)

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Last modified February 17 03:37 EST 2012. Contains 205978 sequences.