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A156816
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Decimal expansion of the positive root of the equation 13x^4 - 7x^2 - 581 = 0.
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0
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2, 6, 3, 8, 1, 5, 8, 5, 3, 0, 3, 4, 1, 7, 4, 0, 8, 6, 8, 4, 3, 0, 3, 0, 7, 5, 6, 6, 7, 4, 4, 4, 1, 3, 0, 4, 8, 8, 8, 0, 5, 0, 2, 2, 0, 1, 0, 3, 1, 8, 3, 5, 9, 7, 3, 7, 0, 7, 8, 7, 0, 6, 0, 7, 7, 6, 9, 6, 3, 2, 1, 9, 7, 0, 7, 3, 5, 5, 9, 5, 9, 8, 8, 9, 3, 2, 0, 0, 5, 1, 8, 9, 0, 0, 0, 9, 8, 3, 3, 5, 2, 4, 2, 1, 2
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OFFSET
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1,1
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COMMENTS
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This constant approximates the connective constant of the square lattice, which is known only numerically, but "no derivation or explanation of this quartic polynomial is known, and later evidence has raised doubts about its validity" [Bauerschmidt et al, 2012, p. 4]. - Andrey Zabolotskiy, Dec 26 2018
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REFERENCES
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N. Madras and G. Slade, The Self-Avoiding Walk (Boston: Birkhauser), 1993.
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LINKS
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Table of n, a(n) for n=1..105.
Roland Bauerschmidt, Hugo Duminil-Copin, Jesse Goodman, Gordon Slade, Lectures on Self-Avoiding Walks, arXiv:1206.2092 [math.PR], 2012.
M. Bousquet-Mélou, A. J. Guttmann and I. Jensen, Self-avoiding walks crossing a square, arXiv:cond-mat/0506341, 2005.
Pierre-Louis Giscard, Que sait-on compter sur un graphe. Partie 3 (in French), Images des Mathématiques, CNRS, 2020.
Jesper Lykke Jacobsen, Christian R. Scullard, Anthony J. Guttmann, On the growth constant for square-lattice self-avoiding walks, J. Phys. A: Math. Theor., 49 (2016), 494004; arXiv:1607.02984 [cond-mat.stat-mech], 2016.
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FORMULA
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x = sqrt(7/26 + sqrt(30261)/26).
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EXAMPLE
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x = 2.63815853034174086843...
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MATHEMATICA
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RealDigits[Sqrt[1/26*(7+Sqrt[30261])], 10, 120][[1]] (* Harvey P. Dale, Nov 22 2014 *)
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PROG
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(PARI) polrootsreal(13*x^4-7*x^2-581)[2] \\ Charles R Greathouse IV, Apr 16 2014
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CROSSREFS
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Cf. A001411, A002931, A179260, A249776.
Sequence in context: A176014 A011447 A076041 * A021383 A296456 A256592
Adjacent sequences: A156813 A156814 A156815 * A156817 A156818 A156819
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KEYWORD
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cons,nonn
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AUTHOR
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Zak Seidov, Feb 16 2009
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STATUS
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approved
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