1,2

Also decimal expansion of the positive root of (x+1)^n - x^(2n). (x+1)^n - x^2n = 0 has only two real roots x1 = -(sqrt(5)-1)/2 = -.618033988749894848204586834... x2 = (sqrt(5)+1)/2 = 1.618033988749894848204586834... for all n > 0 - Cino Hilliard (hillcino368(AT)gmail.com), May 27 2004

The golden ratio phi is the most irrational among irrational numbers; its successive continued fraction convergents F(n+1)/F(n) are the slowest to approximate to its actual value. (I. Stewart, in 'Nature's Numbers', Basic Books 1997.) - Lekraj Beedassy (blekraj(AT)yahoo.com), Jan 21 2005

M. Berg, Phi, the golden ratio (to 4599 decimal places), and Fibonacci numbers, Fib. Quart., 4 (1961), 157-162.

R. A. Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific, River Edge NJ 1997.

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.2.

M. Gardner, The Second Scientific American Book Of Mathematical Puzzles and Diversions, "Phi:The Golden Ratio", Chapter 8, Simon & Schuster NY 1961.

H. E. Huntley, The Divine Proportion, Dover NY 1970.

M. Livio, The Golden Ratio, Broadway Books, NY, 2002.

S. Olsen, The Golden Section, Walker & Co. NY 2006.

H. Walser, The Golden Section, Math. Assoc. of Amer. Washington DC 2001.

C. J. Willard, Le nombre d'or, Magnard Paris 1987.

Robert G. Wilson v, Table of n, a(n) for n=1..100000

John Baez, This week's finds in mathematical physics, Week 203

A. Camus College Team, Le nombre d'or

T. Eveilleau, Le nombre d'or(Text in French)

Gutenberg Project, The golden ratio to 20000 places [link may be broken?]

Heartbeat200.com, Introduction to The Golden Proportion

ICON Project, The golden ratio to 50000 places

R. Knott, Fibonacci numbers and the golden section

E. Levin, The Golden Proportion

Mathematical Database, Poster, The Golden Ratio

Meiner, Phi:The Golden Number

D. Merrill, Fib-Phi Link Page

D. Merrill, Golden ratio to 1000000 digits

J. C. Michel, Le nombre d'or

J. J. O'Connor & E.F.Robertson, The Golden ratio

S. Plouffe, Plouffe's Inverter, The golden ratio to 10 million digits

S. Plouffe, The golden ratio:(1+sqrt(5))/2 to 20000 places

F. Richman, Fibonacci sequence with multiprecision Java, Successive approximations to phi from ratios of consecutive Fibonacci numbers

E. F. Schubert, The Fibonacci series

A. M. Selvam, Golden mean and self-similar,fractal geometrical structures in nature

M. R. Watkins, The "Golden Mean" in number theory

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Silver Ratio

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Wikipedia, Golden mean

1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391138...

RealDigits[(1 + Sqrt[5])/2, 10, 130] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 02 2006

RealDigits[ Exp[ ArcSinh[1/2]], 10, 111][[1]] (* Robert G. Wilson v (rgwv@rgwv.com), Mar 01 2008 *)

Cf. A000012.

Adjacent sequences: A001619 A001620 A001621 this_sequence A001623 A001624 A001625

Sequence in context: A082830 A046902 A094214 this_sequence A021622 A073228 A011490

cons,nonn,nice,easy

njas

Additional links contributed by Lekraj Beedassy (blekraj(AT)yahoo.com), Dec 23 2003

More terms from Gabriel Cunningham (gcasey(AT)mit.edu), Oct 24 2004

More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 02 2006