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

Author

Zak Seidov, Feb 16 2009

Keywords

• cons
• nonn

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

Examples

			x = 2.63815853034174086843...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.10, p. 331.
• 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.
• Index entries for algebraic numbers, degree 4.

Crossrefs

Cf. A001411, A002931, A179260, A249776.

Programs

Mathematica

RealDigits[Sqrt[1/26*(7+Sqrt[30261])],10,120][[1]] (* Harvey P. Dale, Nov 22 2014 *)

PARI

polrootsreal(13*x^4-7*x^2-581)[2] \\ Charles R Greathouse IV, Apr 16 2014

Formula

x = sqrt(7/26 + sqrt(30261)/26).