

A136319


Decimal expansion of [phi, phi, ...] = (phi + sqrt[phi^2 + 4])/2.


4



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, 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, 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
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OFFSET

1,1


COMMENTS

A geometric realization of this number is the ratio of length to width of a metagolden rectangle. See A188635 for details and continued fraction.  Clark Kimberling, Apr 06 2011
This number is the asymptotic limit of the ratio of consecutive terms of the sequence of the number of Khalimskycontinuous functions with fourpoint codomain. See the FORMULA section of A131935 for details. (Cf. Samieinia 2010.)  Geoffrey Caveney, Apr 17 2014
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
Calling this number mu, log(mu) = asinh(phi/2).  Geoffrey Caveney, Apr 21 2014


LINKS

Table of n, a(n) for n=1..100.
R. Evans, H. Hollmann, C. Krattenthaler and Q. Xiang, Gauss sums, Jacobi sums, and pranks of cyclic difference sets, J. Combin. Theory Ser. A, 87.1 (1999), 74119.
Shiva Samieinia, Digital straight line segments and curves. Licentiate Thesis. Stockholm University, Department of Mathematics, Report 2007:6.
Shiva Samieinia, The number of Khalimskycontinuous functions on intervals, Rocky Mountain J. Math., 40.5 (2010), 16671687.
Eric Weisstein's World of Mathematics, Silver Ratio
Wikipedia, Silver ratio


FORMULA

(phi + sqrt(phi^2 + 4))/2.
Also, (1/4)*(1 + sqrt(5) + sqrt(H)), where H = 22 + 2*sqrt(5). (corrected by Jonathan Sondow, Apr 18 2014).
phi*(1 + sqrt(7  2*sqrt(5)))/2.  Geoffrey Caveney, Apr 19 2014
exp(asinh(cos(Pi/5))).  Geoffrey Caveney, Apr 22 2014
cos(Pi/5) + sqrt(1+cos(Pi/5)^2).  Geoffrey Caveney, Apr 23 2014


MAPLE

Digits:=100: evalf((1+sqrt(5))*(1+sqrt(72*sqrt(5)))/4); # Wesley Ivan Hurt, Apr 22 2014


MATHEMATICA

(GoldenRatio + Sqrt[GoldenRatio^2 + 4])/2


CROSSREFS

Cf. A014176, A188635.
Sequence in context: A019693 A007493 A262177 * A176057 A152566 A021481
Adjacent sequences: A136316 A136317 A136318 * A136320 A136321 A136322


KEYWORD

cons,nonn


AUTHOR

Ryan Tavenner (tavs(AT)pacbell.net), Mar 24 2008


STATUS

approved



