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, Jan 21 2005

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

Golden ratio=2*cos(pi*1/5)=e^(i*pi*1/5)+e^(-i*pi*1/5). [From Eric Desbiaux, Mar 19 2010]

Let t=golden ratio.  The lesser sqrt(5)-contraction rectangle has shape t-1, and the greater sqrt(5)-contraction rectangle has shape t.  For definitions of shape and contraction rectangles, see A188739.  [From Clark Kimberling, Apr 16 2011]

The golden ratio (often denoted by phi or tau) is the shape (i.e., length/width) of the golden rectangle, which has the special property that removal of a square from one end leaves a rectangle of the same shape as the original rectangle.  Analogously, removals of certain isosceles triangles characterize side-golden and angle-golden triangles.  Repeated removals in these configurations result in infinite partitions of golden rectangles and triangles into squares or isosceles triangles so as to match the continued fraction, [1,1,1,1,1,...] of tau.  For the special shape of rectangle which partitions into golden rectangles so as to match the continued fraction [tau, tau, tau, ...], see A188635.  For other rectangular shapes which depend on tau, see A189970, A190177, A190179, A180182.  For triangular shapes which depend on tau, see A152149 and A188594; for tetrahedral, see A178988. [From Clark Kimberling, May 06 2011]

Given a pentagon ABCDE, 1/(phi)^2 <= (A*C^2 +C*E^2 +E*B^2 +B*D^2 +D*A^2) / (A*B^2 +B*C^2 +C*D^2 +D*E^2 +E*A^2) <= (phi)^2 [From Seiichi Kirikami, Aug 18 2011]

If a triangle has sides whose lengths form a geometric progression in the ratio of 1:r:r^2 then the triangle inequality condition requires that r be in the range 1/phi < r < phi. - Frank M Jackson, Oct 12 2011

The graphs of x-y=1 and xy=1 meet at (tau,1/tau).  [From Clark Kimberling, Oct 19 2011]

Also decimal expansion of the first root of x^sqrt(x+1) = sqrt(x+1)^x. [From Michel Lagneau, Dec 02 2011].

Also decimal expansion of the  root of (1/x)^(1/sqrt(x+1)) = (1/sqrt(x+1))^(1/x).  [From Michel Lagneau, Apr 17 2012].

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.

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, Aug 27 2008]

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

L. B. W. Jolley, The summation of series, Dover (1961).

M. Livio, The Golden Ratio, Broadway Books, NY, 2002. [see the review by G. Markowsky in the links field]

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

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

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

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

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

Gutenberg Project, The golden ratio to 20000 places

ICON Project, The golden ratio to 50000 places

R. Knott, Fibonacci numbers and the golden section

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

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

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

J. Sondow, Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers, Diophantine Analysis and Related Fields 2011 - AIP Conference Proceedings, vol. 1385, pp. 97-100.

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

Eric Weisstein's World of Mathematics, Golden Ratio

Eric Weisstein's World of Mathematics, Silver Ratio

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

Wikipedia, Golden mean

Comments from Hieronymus Fischer, 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 minimal 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)

The solutions to x-x^(-1)=floor(x) are determined by x=1/2*(m+sqrt(m^2+4)), m>=1; x=phi for m=1. In terms of continued fractions the solutions can be described by x=[m;m,m,m,...], where m=1 for x=phi, and m=2 for the silver ratio A014176, and m=3 for the bronze ratio A098316. - Hieronymus Fischer, October 20 2010

Sum_{n>=1} x^n/n^2 = Pi^2/10-(log(2)*sin(Pi/10))^2 where x= 2*sin(Pi/10) = this constant here. [Jolley eq 360d]

1.6180339887498948482045868343656381177203091798057628621...

RealDigits[(1 + Sqrt[5])/2, 10, 130] - (* Stefan Steinerberger, Apr 02 2006*)

RealDigits[ Exp[ ArcSinh[1/2]], 10, 111][[1]] (* Robert G. Wilson v, 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)); }

/* Harry J. Smith, Apr 19 2009 */

(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})

/* Michael Porter, Oct 24 2009 */

Cf. A000012, A000032, A000045, A006497, A080039, A104457, A188635, A192222, A192223.

A145996 [From Artur Jasinski, Oct 26 2008].

A139339, A197762 (hyperbola intersections) [From Clark Kimberling, Oct 19 2011]

Cf. A002163, A094874, A134973.

Sequence in context: A143019 A156921 A094214 * A186099 A021622 A073228

Adjacent sequences:  A001619 A001620 A001621 * A001623 A001624 A001625

cons,nonn,nice,easy

N. J. A. Sloane.

Additional links contributed by Lekraj Beedassy, Dec 23 2003

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

More terms from Stefan Steinerberger, Apr 02 2006

Broken URL to Project Gutenberg replaced by Georg Fischer, Jan 03 2009

Corrected PARI program, had -n Harry J. Smith, May 17 2009

approved