A370088 Decimal expansion of the two-dimensional backbone constant.

3, 5, 6, 6, 6, 6, 8, 3, 6, 7, 1, 2, 8, 8, 9, 5, 8, 2, 8, 3, 7, 3, 0, 7, 3, 8, 1, 0, 0, 1, 2, 6, 6, 2, 6, 9, 9, 0, 3, 8, 7, 0, 1, 5, 3, 4, 0, 7, 6, 2, 4, 4, 1, 3, 9, 9, 0, 6, 0, 9, 7, 3, 7, 6, 3, 7, 3, 6, 1, 3, 8, 4, 2, 0, 8, 8, 8, 5, 5, 4, 8, 5, 1, 9, 6, 7, 2

Offset

0,1

Comments

This constant is the negative of the exponent of the growth rate of the probability of Bernoulli percolation on the 2-dimensional triangular lattice at criticality (p = 1/2). It is transcendental (Theorem 1.2 in Nolin, Qian, Sun, & Zhuang).

Links

Table of n, a(n) for n=0..86.

Pierre Nolin, Wei Qian, Xin Sun, and Zijie Zhuang, Backbone exponent for two-dimensional percolation, arXiv:2309.05050 [math.PR], 2023-2024.

Leila Sloman, Maze proof establishes a 'backbone' for statistical mechanics, Quanta Magazine (2024).

Index entries for transcendental numbers

Formula

This is the unique constant 1/4 < x < 2/3 with sqrt(36*x+3)/4 + sin(2*Pi*sqrt(12*x+1)/3) = 0.

Example

0.35666683671288958283730738100126626990387015340762441399060973763736138420....

Mathematica

RealDigits[x /. FindRoot[Sqrt[36*x + 3]/4 + Sin[2*Pi*Sqrt[12*x + 1]/3], {x, 1/2}, WorkingPrecision -> 105]][[1]] (* Amiram Eldar, Mar 03 2026 *)

Prog

(PARI) t=sqrt(3)/4; u=2*Pi/3; solve(x=.3, .4, my(s=sqrt(12*x+1)); sin(s*u)+s*t)
(PARI) t=sqrt(3)/4; u=2*Pi/3; (solve(s=2.2, 2.3, sin(s*u)+s*t)^2-1)/12 \\ Charles R Greathouse IV, Apr 26 2026

Crossrefs

Cf. A003197, A003198, A003202, A006809.

Sequence in context: A224831 A281591 A267884 * A077859 A342269 A364751

Adjacent sequences: A370085 A370086 A370087 * A370089 A370090 A370091

Keyword

cons,nonn

Author

Charles R Greathouse IV, Feb 08 2024

Status

approved