A176394 Decimal expansion of 3+2*sqrt(3).

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, 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, 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

Offset: 1

Author

Klaus Brockhaus, Apr 16 2010

Keywords

• cons
• nonn

Comments

Continued fraction expansion of 3+2*sqrt(3) is A010696 preceded by 6.

a(n) = A010469(n) for n > 1.

Largest radius of three circles tangent to a circle of radius 1. - Charles R Greathouse IV, Jan 14 2013

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

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

Examples

			3+2*sqrt(3) = 6.46410161513775458705...
		

Links

• Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
• John Baez, Black Holes and The Golden Ratio, Mar 01 2013.
• Index entries for algebraic numbers, degree 2.

Crossrefs

Cf. A002194 (decimal expansion of sqrt(3)), A010469 (decimal expansion of sqrt(12)), A010696 (repeat 2, 6).

Programs

Mathematica

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 *)

PARI

3+2*sqrt(3) \\ Charles R Greathouse IV, Jan 14 2013

Formula

Equals Sum_{n>=1} (sqrt(3)/2)^n = (sqrt(3)/2)/(1 - (sqrt(3)/2)). - Fred Daniel Kline, Mar 03 2014