%I #48 Jun 20 2021 11:43:32
%S 2,0,9,5,2,9,3,9,8,5,2,2,3,9,1,4,4,9,2,7,4,6,8,1,6,7,1,8,8,6,6,2,8,2,
%T 5,8,3,1,6,6,4,1,1,5,2,7,5,7,3,8,3,6,8,9,4,4,0,4,7,7,5,5,4,6,6,5,4,5,
%U 3,7,8,5,0,7,6,3,9,7,8,6,1,3,7,5,2,1,8,3,6,1,4,3,0,7,4,7,1,3,5,3
%N Decimal expansion of [phi, phi, ...] = (phi + sqrt(phi^2 + 4))/2.
%C A geometric realization of this number is the ratio of length to width of a meta-golden rectangle. See A188635 for details and continued fraction. - _Clark Kimberling_, Apr 06 2011
%C This number is the asymptotic limit of the ratio of consecutive terms of the sequence of the number of Khalimsky-continuous functions with four-point codomain. See the FORMULA section of A131935 for details. (Cf. Samieinia 2010.) - _Geoffrey Caveney_, Apr 17 2014
%C This number is the largest zero of the polynomial z^4 - z^3 - 3*z^2 + z + 1. (Cf. Evans, Hollmann, Krattenthaler and Xiang 1999, p. 107.) - _Geoffrey Caveney_, Apr 17 2014
%C Calling this number mu, log(mu) = arcsinh(phi/2). - _Geoffrey Caveney_, Apr 21 2014
%H R. Evans, H. Hollmann, C. Krattenthaler and Q. Xiang, <a href="http://dx.doi.org/10.1006/jcta.1998.2950">Gauss sums, Jacobi sums, and p-ranks of cyclic difference sets</a>, J. Combin. Theory Ser. A, 87.1 (1999), 74-119.
%H Shiva Samieinia, <a href="http://www.math.su.se/reports/2007/6/">Digital straight line segments and curves</a>. Licentiate Thesis. Stockholm University, Department of Mathematics, Report 2007:6.
%H Shiva Samieinia, <a href="http://dx.doi.org/10.1216/RMJ-2010-40-5-1667">The number of Khalimsky-continuous functions on intervals</a>, Rocky Mountain J. Math., 40.5 (2010), 1667-1687.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SilverRatio.html">Silver Ratio</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/silver_ratio">Silver ratio</a>
%F (phi + sqrt(phi^2 + 4))/2.
%F Also, (1/4)*(1 + sqrt(5) + sqrt(H)), where H = 22 + 2*sqrt(5). (corrected by _Jonathan Sondow_, Apr 18 2014)
%F phi*(1 + sqrt(7 - 2*sqrt(5)))/2. - _Geoffrey Caveney_, Apr 19 2014
%F exp(arcsinh(cos(Pi/5))). - _Geoffrey Caveney_, Apr 22 2014
%F cos(Pi/5) + sqrt(1+cos(Pi/5)^2). - _Geoffrey Caveney_, Apr 23 2014
%p Digits:=100: evalf((1+sqrt(5))*(1+sqrt(7-2*sqrt(5)))/4); # _Wesley Ivan Hurt_, Apr 22 2014
%t RealDigits[(GoldenRatio+Sqrt[GoldenRatio^2+4])/2,10,120][[1]] (* _Harvey P. Dale_, Jun 20 2021 *)
%Y Cf. A001622, A014176, A188635.
%K cons,nonn
%O 1,1
%A Ryan Tavenner (tavs(AT)pacbell.net), Mar 24 2008
%E Previous Mathematica program corrected and replaced by _Harvey P. Dale_, Jun 20 2021