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

GoldenRatio=Hypergeometric2F1[1/5, 4/5, 1/2, 3/4]=2*Cos[(3/5)*ArcSin[Sqrt[3/4]]] [From Artur Jasinski (grafix(AT)csl.pl), Oct 26 2008]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

S. 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.

M. Gardner, Weird Water and Fuzzy Logic: More Notes of a Fringe Watcher, "The Cult of the Golden Ratio", Chapter 9, Prometheus Books, 1996, pages 90-97. [From William Rex Marshall (w.r.marshall(AT)actrix.co.nz), Aug 27 2008]

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

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

G. Markowsky, Misconceptions About the Golden Ratio, College Mathematics Journal, 23:1 (January 1992), 2-19. [From William Rex Marshall (w.r.marshall(AT)actrix.co.nz), Aug 27 2008]

G. Markowsky, Book review: The Golden Ratio, Notices of the AMS, 52:3 (March 2005), 344-347. [From William Rex Marshall (w.r.marshall(AT)actrix.co.nz), Aug 27 2008]

Comments from Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jan 02 2009 (Start): The fractional part of phi^n equals phi^(-n), if n odd. For even n, the fractional part of phi^n is equal to 1-phi^(-n).

General formula: Provided x>1 suffices x-x^(-1)=floor(x), where x=phi for this sequence, then

for odd n: x^n-x^(-n)=floor(x^n), hence fract(x^n)=x^(-n),

for even n: x^n+x^(-n)=ceiling(x^n), hence fract(x^n)=1-x^(-n),

for all n>0: x^n+(-x)^(-n)=nint(x^n).

x=phi is the minmal solution to x-x^(-1)=floor(x) (where floor(x)=1 in this case).

Other examples of constants x satisfying the relation x-x^(-1)=floor(x) include A014176 (the silver ratio: where floor(x)=2) and A098316 (the "bronze" ratio: where floor(x)=3). (End)

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(AT)rgwv.com), Mar 01 2008 *)

(PARI) { default(realprecision, 20080); x=(1+sqrt(5))/2; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b001622.txt", n, " ", d)); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 19 2009]

Contribution from Michael Porter (michael_b_porter(AT)yahoo.com), Oct 24 2009: (Start)

(PARI) /* Digit-by-digit method */

/* write it as 0.5+sqrt(1.25) and start at hundredths digit */

r=11; x=400; print(1); print(6);

for(digits=1, 110, {d=0; while((20*r+d)*d <= x, d++);

d--; /* while loop overshoots correct digit */

print(d); x=100*(x-(20*r+d)*d); r=10*r+d}) (End)

Cf. A000012.

Cf. A104457.

A145996 [From Artur Jasinski (grafix(AT)csl.pl), Oct 26 2008]

Cf. A000032, A006497, A080039.

Sequence in context: A143019 A156921 A094214 this_sequence A021622 A073228 A145314

Adjacent sequences: A001619 A001620 A001621 this_sequence A001623 A001624 A001625

cons,nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).

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

Broken URL to Project Gutenberg replaced by Georg Fischer (Georg.Fischer(AT)T-Online.de), Jan 03 2009

Corrected PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 17 2009