A156816 Decimal expansion of the positive root of the equation 13x^4 - 7x^2 - 581 = 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

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

Table of n, a(n) for n=1..105.

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.

Index entries for algebraic numbers, degree 4

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

Cf. A001411, A002931, A179260, A249776.

Sequence in context: A176014 A011447 A076041 * A021383 A296456 A256592

Adjacent sequences: A156813 A156814 A156815 * A156817 A156818 A156819

Keyword

cons,nonn

Author

Zak Seidov, Feb 16 2009

Status

approved