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%I #38 Mar 14 2020 06:56:19
%S 6,4,6,4,1,0,1,6,1,5,1,3,7,7,5,4,5,8,7,0,5,4,8,9,2,6,8,3,0,1,1,7,4,4,
%T 7,3,3,8,8,5,6,1,0,5,0,7,6,2,0,7,6,1,2,5,6,1,1,1,6,1,3,9,5,8,9,0,3,8,
%U 6,6,0,3,3,8,1,7,6,0,0,0,7,4,1,6,2,2,9,2,3,7,3,5,1,4,4,9,7,1,5,1,3,5,1,2,5
%N Decimal expansion of 3+2*sqrt(3).
%C Continued fraction expansion of 3+2*sqrt(3) is A010696 preceded by 6.
%C a(n) = A010469(n) for n > 1.
%C Largest radius of three circles tangent to a circle of radius 1. - _Charles R Greathouse IV_, Jan 14 2013
%C For a spinning black hole the phase transition to positive specific heat happens at a point governed by 2*sqrt(3)-3 (according to a discussion on John Baez's blog), and not at the golden ratio as claimed by Paul Davis. - _Peter Luschny_, Mar 02 2013
%C In particular: a black hole with J > (2*sqrt(3)-3) Gm^2/c has positive specific heat, and negative specific heat if J is less, where J is its angular momentum, m is its mass, G is the gravitational constant, and c is the speed of light. For a solar mass black hole, this is 4.08 * 10^41 joule-seconds or a rotation every 1.61 days with the sun's inertia. - _Charles R Greathouse IV_, Sep 20 2013
%H Daniel Starodubtsev, <a href="/A176394/b176394.txt">Table of n, a(n) for n = 1..10000</a>
%H John Baez, <a href="http://johncarlosbaez.wordpress.com/2013/02/28/black-holes-and-the-golden-ratio/">Black Holes and The Golden Ratio</a>, Mar 01 2013.
%F Equals Sum_{n>=1} (sqrt(3)/2)^n = (sqrt(3)/2)/(1 - (sqrt(3)/2)). - _Fred Daniel Kline_, Mar 03 2014
%e 3+2*sqrt(3) = 6.46410161513775458705...
%t Circs[n_] := With[{r = Sin[Pi/n]/(1 - Sin[Pi/n])}, Graphics[Append[Table[Circle[(r + 1) {Sin[2 Pi k/n], Cos[2 Pi k/n]}, r], {k, n}], {Blue, Circle[{0, 0}, 1]}]]]; Circs[3] (* _Charles R Greathouse IV_, Jan 14 2013 *)
%o (PARI) 3+2*sqrt(3) \\ _Charles R Greathouse IV_, Jan 14 2013
%Y Cf. A002194 (decimal expansion of sqrt(3)), A010469 (decimal expansion of sqrt(12)), A010696 (repeat 2, 6).
%K cons,nonn
%O 1,1
%A _Klaus Brockhaus_, Apr 16 2010
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