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%I #46 Dec 09 2021 11:17:33
%S 4,4,7,2,1,3,5,9,5,4,9,9,9,5,7,9,3,9,2,8,1,8,3,4,7,3,3,7,4,6,2,5,5,2,
%T 4,7,0,8,8,1,2,3,6,7,1,9,2,2,3,0,5,1,4,4,8,5,4,1,7,9,4,4,9,0,8,2,1,0,
%U 4,1,8,5,1,2,7,5,6,0,9,7,9,8,8,2,8,8,2,8,8,1,6,7,5,7,5,6,4,5,4,9,9,3,9,0,1
%N Decimal expansion of 1/sqrt(5).
%C This number is the cosine of the central angle of a regular icosahedron; see A105199 for the angle itself. - _Clark Kimberling_, Feb 10 2009
%C Largest radius of ten circles tangent to a circle of radius 1. - _Charles R Greathouse IV_, Jan 14 2013
%H Ivan Panchenko, <a href="/A020762/b020762.txt">Table of n, a(n) for n = 0..1000</a>
%H Hideyuki Ohtsuka, <a href="https://www.fq.math.ca/Problems/ElemProbSolnMay14.pdf">Problem B-1148</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 52, No. 2 (2014), p. 179; <a href="https://www.fq.math.ca/Problems/ElemProbSolnMay15.pdf">The Exact Value of an Infinite Series</a>, Solution to B-1148, ibid., Vol. 53, No. 2 (2015), pp. 183-184.
%F Equals cos(arctan(2)). - _Clark Kimberling_, Feb 10 2009
%F Equals lim_{n -> infinity} A000045(n)/A000032(n). - _Bruno Berselli_, Jan 22 2018
%F From _Christian Katzmann_, Mar 19 2018: (Start)
%F Equals Sum_{n>=0} (2*n)!/(n!^2*3^(2*n+1)).
%F Equals Sum_{n>=0} 5*(2*n+1)!/(n!^2*3^(2*n+3)). (End)
%F Equals A010476/10. - _R. J. Mathar_, Jan 14 2021
%F 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
%e 0.447213595499957939281834733746255247088123671922305144854179449082104...
%t RealDigits[5^(-1/2), 10, 150] (* _Stefan Steinerberger_, Apr 08 2006 *)
%t Circs[n_] := With[{r = Sin[Pi/n]/(1 - Sin[Pi/n])}, Graphics[Append[
%t Table[Circle[(r + 1) {Sin[2 Pi k/n], Cos[2 Pi k/n]}, r], {k, n}],
%t {Blue, Circle[{0, 0}, 1]}]]]
%t Circs[10] (* _Charles R Greathouse IV_, Jan 14 2013 *)
%o (PARI) 1/sqrt(5) \\ _Charles R Greathouse IV_, Jan 14 2013
%Y Cf. A000032, A000045, A010476, A001566, A058635, A105199.
%K nonn,cons
%O 0,1
%A _N. J. A. Sloane_
%E More terms from _Stefan Steinerberger_, Apr 08 2006
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