OFFSET
0,1
COMMENTS
This number is the cosine of the central angle of a regular icosahedron; see A105199 for the angle itself. - Clark Kimberling, Feb 10 2009
Largest radius of ten circles tangent to a circle of radius 1. - Charles R Greathouse IV, Jan 14 2013
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..1000
Hideyuki Ohtsuka, Problem B-1148, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 52, No. 2 (2014), p. 179; The Exact Value of an Infinite Series, Solution to B-1148, ibid., Vol. 53, No. 2 (2015), pp. 183-184.
FORMULA
Equals cos(arctan(2)). - Clark Kimberling, Feb 10 2009
From Christian Katzmann, Mar 19 2018: (Start)
Equals Sum_{n>=0} (2*n)!/(n!^2*3^(2*n+1)).
Equals Sum_{n>=0} 5*(2*n+1)!/(n!^2*3^(2*n+3)). (End)
Equals A010476/10. - R. J. Mathar, Jan 14 2021
Equals Sum_{k>=1} F(2^(k-1))/(L(2^k)+1) = Sum_{k>=0} A058635(k)/(A001566(k)+1), where F(k) = A000045(k) is the k-th Fibonacci number and L(k) = A000032(k) is the k-th Lucas number (Ohtsuka, 2014). - Amiram Eldar, Dec 09 2021
EXAMPLE
0.447213595499957939281834733746255247088123671922305144854179449082104...
MATHEMATICA
RealDigits[5^(-1/2), 10, 150] (* Stefan Steinerberger, Apr 08 2006 *)
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[10] (* Charles R Greathouse IV, Jan 14 2013 *)
PROG
(PARI) 1/sqrt(5) \\ Charles R Greathouse IV, Jan 14 2013
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
EXTENSIONS
More terms from Stefan Steinerberger, Apr 08 2006
STATUS
approved