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A156816 Decimal expansion of the positive root of the equation 13x^4 - 7x^2 - 581 = 0. 0
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
REFERENCES
N. Madras and G. Slade, The Self-Avoiding Walk (Boston, Birkhauser), 1993.
LINKS
Roland Bauerschmidt, Hugo Duminil-Copin, Jesse Goodman, and 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, and 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.
FORMULA
x = sqrt(7/26 + sqrt(30261)/26).
EXAMPLE
x = 2.63815853034174086843...
MATHEMATICA
RealDigits[Sqrt[1/26*(7+Sqrt[30261])], 10, 120][[1]] (* Harvey P. Dale, Nov 22 2014 *)
PROG
(PARI) polrootsreal(13*x^4-7*x^2-581)[2] \\ Charles R Greathouse IV, Apr 16 2014
CROSSREFS
Sequence in context: A176014 A011447 A076041 * A021383 A296456 A256592
KEYWORD
cons,nonn
AUTHOR
Zak Seidov, Feb 16 2009
STATUS
approved

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Last modified April 22 01:34 EDT 2024. Contains 371887 sequences. (Running on oeis4.)