A136319 Decimal expansion of [phi, phi, ...] = (phi + sqrt(phi^2 + 4))/2.

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

Offset: 1

Author

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

Keywords

• cons
• nonn

Comments

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

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

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) = arcsinh(phi/2). - Geoffrey Caveney, Apr 21 2014

Links

• R. Evans, H. Hollmann, C. Krattenthaler and Q. Xiang, Gauss sums, Jacobi sums, and p-ranks of cyclic difference sets, J. Combin. Theory Ser. A, 87.1 (1999), 74-119.
• Shiva Samieinia, Digital straight line segments and curves. Licentiate Thesis. Stockholm University, Department of Mathematics, Report 2007:6.
• Shiva Samieinia, The number of Khalimsky-continuous functions on intervals, Rocky Mountain J. Math., 40.5 (2010), 1667-1687.
• Eric Weisstein's World of Mathematics, Silver Ratio
• Wikipedia, Silver ratio

Crossrefs

Cf. A001622, A014176, A188635.

Programs

Maple

Digits:=100: evalf((1+sqrt(5))*(1+sqrt(7-2*sqrt(5)))/4); # Wesley Ivan Hurt, Apr 22 2014

Mathematica

RealDigits[(GoldenRatio+Sqrt[GoldenRatio^2+4])/2,10,120][[1]] (* Harvey P. Dale, Jun 20 2021 *)

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(arcsinh(cos(Pi/5))). - Geoffrey Caveney, Apr 22 2014

cos(Pi/5) + sqrt(1+cos(Pi/5)^2). - Geoffrey Caveney, Apr 23 2014

Extensions

Previous Mathematica program corrected and replaced by Harvey P. Dale, Jun 20 2021