Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #11 Feb 11 2024 09:16:50
%S 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,
%T 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,
%U 1,3,8,4,2,0,8,8,8,5,5,4,8,5,1,9,6,7,2
%N Decimal expansion of the two-dimensional backbone constant.
%C 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).
%H Pierre Nolin, Wei Qian, Xin Sun, and Zijie Zhuang, <a href="https://arxiv.org/abs/2309.05050">Backbone exponent for two-dimensional percolation</a>, arXiv preprint (2023). arXiv:2309.05050 [math.PR], 2023-2024.
%H Leila Sloman, <a href="https://www.quantamagazine.org/maze-proof-establishes-a-backbone-for-statistical-mechanics-20240207/">Maze proof establishes a 'backbone' for statistical mechanics</a>, Quanta Magazine (2024).
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F 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.
%e 0.35666683671288958283730738100126626990387015340762441399060973763736138420....
%o (PARI) t=sqrt(3)/4; u=2*Pi/3; solve(x=.3,.4, my(s=sqrt(12*x+1)); sin(s*u)+s*t)
%Y Cf. A003197, A003198, A003202, A006809.
%K cons,nonn
%O 0,1
%A _Charles R Greathouse IV_, Feb 08 2024