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A001622 Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2.
(Formerly M4046 N1679)

%I M4046 N1679

%S 1,6,1,8,0,3,3,9,8,8,7,4,9,8,9,4,8,4,8,2,0,4,5,8,6,8,3,4,3,6,5,6,3,8,

%T 1,1,7,7,2,0,3,0,9,1,7,9,8,0,5,7,6,2,8,6,2,1,3,5,4,4,8,6,2,2,7,0,5,2,

%U 6,0,4,6,2,8,1,8,9,0,2,4,4,9,7,0,7,2,0,7,2,0,4,1,8,9,3,9,1,1,3,7,4,8,4,7,5

%N Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2.

%C 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 and x2 = (sqrt(5)+1)/2 for all n > 0. - _Cino Hilliard_, May 27 2004

%C 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

%C 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. - _Clark Kimberling_, Apr 16 2011

%C 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. - _Clark Kimberling_, May 06 2011

%C 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. - _Seiichi Kirikami_, Aug 18 2011

%C 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

%C The graphs of x-y=1 and x*y=1 meet at (tau,1/tau). - _Clark Kimberling_, Oct 19 2011

%C Also decimal expansion of the first root of x^sqrt(x+1) = sqrt(x+1)^x. - _Michel Lagneau_, Dec 02 2011

%C Also decimal expansion of the root of (1/x)^(1/sqrt(x+1)) = (1/sqrt(x+1))^(1/x). - _Michel Lagneau_, Apr 17 2012

%C This is the case n=5 of (Gamma(1/n)/Gamma(3/n))*(Gamma((n-1)/n)/Gamma((n-3)/n)): (1+sqrt(5))/2 = (Gamma(1/5)/Gamma(3/5))*(Gamma(4/5)/Gamma(2/5)). - _Bruno Berselli_, Dec 14 2012

%C Also decimal expansion of the only number x>1 such that (x^x)^(x^x)= (x^(x^x))^x = x^((x^x)^x). - _Jaroslav Krizek_, Feb 01 2014

%C For n >= 1, round(phi^prime(n)) == 1 (mod prime(n)) and, for n >= 3, round(phi^prime(n)) == 1 (mod 2*prime(n)). - _Vladimir Shevelev_, Mar 21 2014

%C The continuous radical sqrt(1+sqrt(1+sqrt(1+... tends to phi. - _Giovanni Zedda_, Jun 22 2019

%D Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3, Fall 1998, p. 176. Solution published in Vol. 12, No. 1, Winter 2000, pp. 61-62.

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

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

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

%D 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.

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

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

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

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

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

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

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

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

%H Robert G. Wilson v, <a href="/A001622/b001622.txt">Table of n, a(n) for n = 1..100000</a>

%H John Baez, <a href="http://math.ucr.edu/home/baez/week203.html">This week's finds in mathematical physics, Week 203</a>

%H John Baez, <a href="http://math.ucr.edu/home/baez/numbers/">The Rankin Lectures 2008, My Favorite Numbers: 5</a>. [<a href="http://www.youtube.com/watch?v=2oPGmxDua2U">video</a>]

%H M. Berg, <a href="http://www.fq.math.ca/Scanned/4-2/berg.pdf">Phi, the golden ratio (to 4599 decimal places) and Fibonacci numbers</a>, Fib. Quart., 4 (1961), 157-162.

%H Ömür Deveci, Zafer Adıgüzel, and Taha Doğan, <a href="https://doi.org/10.7546/nntdm.2020.26.1.179-190">On the Generalized Fibonacci-circulant-Hurwitz numbers</a>, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 1, 179-190.

%H T. Eveilleau, <a href="http://perso.orange.fr/therese.eveilleau/pages/truc_mat/textes/rectangle_dor.htm">Le nombre d'or</a> (in French)

%H Gutenberg Project, <a href="http://www.gutenberg.org/etext/633">The golden ratio to 20000 places</a>

%H ICON Project, <a href="http://www.cs.arizona.edu/icon/oddsends/phi.htm">The golden ratio to 50000 places</a>

%H Franklin H. J. Kenter, <a href="https://arxiv.org/abs/1712.04856">It's good to be phi: a solution to a problem of Gosper and Knuth</a>, arXiv:1712.04856 [math.HO], 2017.

%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Kimberling/kimber12.html">Lucas Representations of Positive Integers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.

%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/">Fibonacci numbers and the golden section</a>

%H Wolfdieter Lang, <a href="https://www.itp.kit.edu/~wl/EISpub/A333852.pdf">A list of representative simple difference sets of the Singer type for small orders m</a>, Karlsruher Institut für Technologie (Karlsruhe, Germany 2020).

%H S. Litsyn and V. Shevelev, <a href="http://dx.doi.org/10.1142/S1793042105000339">Irrational Factors Satisfying the Little Fermat Theorem</a>, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.

%H G. Markowsky, <a href="http://www.umcs.maine.edu/~markov/GoldenRatio.pdf">Misconceptions About the Golden Ratio</a>, College Mathematics Journal, 23:1 (January 1992), 2-19.

%H G. Markowsky, <a href="http://www.ams.org/notices/200503/rev-markowsky.pdf">Book review: The Golden Ratio</a>, Notices of the AMS, 52:3 (March 2005), 344-347.

%H J. C. Michel, <a href="http://jc.michel.free.fr/nombre_d_or.php">Le nombre d'or</a>

%H J. J. O'Connor & E. F. Robertson, <a href="http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Golden_ratio.html">The Golden ratio</a>

%H Hugo Pfoertner, <a href="/A001622/a001622.txt">1 million digits of phi.</a> Computed using A. J. Yee's y-cruncher.

%H Simon Plouffe, Plouffe's Inverter, <a href="http://www.plouffe.fr/simon/constants/golden.txt">The golden ratio to 10 million digits</a>. [Only announcement, file truncated]

%H Simon Plouffe, <a href="http://web.archive.org/web/20150911212308/http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap46.html">The golden ratio:(1+sqrt(5))/2 to 20000 places</a>.

%H F. Richman, Fibonacci sequence with multiprecision Java, <a href="http://math.fau.edu/Richman/fibjava.htm">Successive approximations to phi from ratios of consecutive Fibonacci numbers</a>

%H Herman P. Robinson, <a href="/A257574/a257574.pdf">The CSR Function</a>, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC35-3 to PC35-4. Annotated and scanned copy.

%H E. F. Schubert, <a href="http://www.rpi.edu/~schubert/Educational-resources/Fibonacci%20series.pdf">The Fibonacci series</a>

%H V. Shevelev, <a href="http://list.seqfan.eu/pipermail/seqfan/2014-March/012737.html">A property of n-bonacci constant</a>, Seqfan (Mar 23 2014)

%H J. Sondow, <a href="http://arxiv.org/abs/1106.4246">Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers</a>, Diophantine Analysis and Related Fields 2011 - AIP Conference Proceedings, vol. 1385, pp. 97-100.

%H M. R. Watkins, <a href="http://www.maths.ex.ac.uk/~mwatkins/zeta/goldenmean.htm">The "Golden Mean" in number theory</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SilverRatio.html">Silver Ratio</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>

%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Golden_ratio">Golden ratio</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Kronecker-Weber_theorem">Kronecker Weber theorem</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Metallic_mean">Metallic mean</a>

%H Alexander J. Yee, <a href="http://www.numberworld.org/y-cruncher">y-cruncher - A Multi-Threaded Pi-Program</a>

%F Equals Hypergeometric2F1([1/5, 4/5], [1/2], 3/4) = 2*cos((3/5)*arcsin(sqrt(3/4))). - _Artur Jasinski_, Oct 26 2008

%F From _Hieronymus Fischer_, Jan 02 2009: (Start)

%F 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).

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

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

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

%F for all n>0: x^n + (-x)^(-n) = round(x^n).

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

%F 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)

%F Equals 2*cos(Pi*1/5) = e^(i*Pi*1/5)+e^(-i*Pi*1/5). - _Eric Desbiaux_, Mar 19 2010

%F 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_, Oct 20 2010

%F 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]

%F phi = 1 + Sum_{k>=1} (-1)^(k-1)/(F(k)*F(k+1)), where F(n) is the n-th Fibonacci number (A000045). Proof. By Catalan's identity, F^2(n) - F(n-1)*F(n+1) = (-1)^(n-1). Therefore,(-1)^(n-1)/(F(n)*F(n+1)) = F(n)/F(n+1) - F(n-1)/F(n). Thus Sum_{k=1..n} (-1)^(k-1)/(F(k)*F(k+1)) = F(n)/F(n+1). If n goes to infinity, this tends to 1/phi = phi - 1. - _Vladimir Shevelev_, Feb 22 2013

%F phi^n = (A000032(n) + A000045(n)*sqrt(5)) / 2. - _Thomas Ordowski_, Jun 09 2013

%F Let P(q) = Product_{k>=1} (1 + q^(2*k-1)) (the g.f. of A000700), then A001622 = exp(Pi/6) * P(exp(-5*Pi)) / P(exp(-Pi)). - _Stephen Beathard_, Oct 06 2013

%F phi = i^(2/5) + i^(-2/5) = ((i^(4/5))+1) / (i^(2/5)) = 2*(i^(2/5) - (sin(Pi/5))i) = 2*(i^(-2/5) + (sin(Pi/5))i). - _Jaroslav Krizek_, Feb 03 2014

%F phi = sqrt(2/(3 - sqrt(5))). This follows from the fact that ((1 + sqrt(5))^2)*(3 - sqrt(5)) = 8, so that ((1 + sqrt(5))/2)^2 = 2/(3 - sqrt(5)). - _Geoffrey Caveney_, Apr 19 2014

%F exp(arcsinh(cos(Pi/2-log(phi)*i))) = exp(arcsinh(sin(log(phi)*i))) = (sqrt(3) + i) / 2. - _Geoffrey Caveney_, Apr 23 2014

%F exp(arcsinh(cos(Pi/3))) = phi. - _Geoffrey Caveney_, Apr 23 2014

%F cos(Pi/3) + sqrt(1 + cos(Pi/3)^2). - _Geoffrey Caveney_, Apr 23 2014

%F 2*phi = z^0 + z^1 - z^2 - z^3 + z^4, where z = exp(2*pi*i/5). See the Wikipedia Kronecker-Weber theorem link. - _Jonathan Sondow_, Apr 24 2014

%F phi = 1/2 + sqrt(1 + (1/2)^2). - _Geoffrey Caveney_, Apr 25 2014

%F Phi is the limiting value of the iteration of x -> sqrt(1+x) on initial value a >= -1. - _Chayim Lowen_, Aug 30 2015

%F a(n) = -10*floor((sqrt(5) + 1)/2*10^(-2 + n)) + floor((sqrt(5) + 1)/2*10^(-1 + n)) for n > 0. - _Mariusz Iwaniuk_, Apr 28 2017

%F From _Isaac Saffold_, Feb 28 2018: (Start)

%F 1 = Sum_{k=0..n} binomial(n, k) / phi^(n+k) for all nonnegative integers n.

%F 1 = Sum_{n>=1} 1 / phi^(2n-1).

%F 1 = Sum_{n>=2} 1 / phi^n.

%F phi = Sum_{n>=1} 1/phi^n. (End)

%F From _Christian Katzmann_, Mar 19 2018: (Start)

%F phi = Sum_{n>=0} (15*(2*n)!+8*n!^2)/(2*n!^2*3^(2*n+2)).

%F phi = 1/2 + Sum_{n>=0} 5*(2*n)!/(2*n!^2*3^(2*n+1)). (End)

%e 1.6180339887498948482045868343656381177203091798057628621...

%p Digits:=1000; evalf((1+sqrt(5))/2); # _Wesley Ivan Hurt_, Nov 01 2013

%t RealDigits[(1 + Sqrt[5])/2, 10, 130] (* _Stefan Steinerberger_, Apr 02 2006 *)

%t RealDigits[ Exp[ ArcSinh[1/2]], 10, 111][[1]] (* _Robert G. Wilson v_, Mar 01 2008 *)

%t RealDigits[GoldenRatio,10,120][[1]] (* _Harvey P. Dale_, Oct 28 2015 *)

%o (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

%o (PARI)

%o /* Digit-by-digit method : write it as 0.5+sqrt(1.25) and start at hundredths digit */

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

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

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

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

%o \\ _Michael B. Porter_, Oct 24 2009

%o (Python)

%o from sympy import S

%o def alst(n): # truncate extra last digit to avoid rounding

%o return list(map(int, str(S.GoldenRatio.n(n+1)).replace(".", "")))[:-1]

%o print(alst(105)) # _Michael S. Branicky_, Jan 06 2021

%Y Cf. A000012, A000032, A000045, A006497, A080039, A104457, A188635, A192222, A192223, A145996, A139339, A197762, A002163, A094874, A134973.

%Y Cf. A102208, A102769, A131595.

%Y Cf. A302973, A303069, A304022.

%K nonn,cons,nice,easy

%O 1,2

%A _N. J. A. Sloane_

%E Additional links contributed by _Lekraj Beedassy_, Dec 23 2003

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

%E More terms from _Stefan Steinerberger_, Apr 02 2006

%E Broken URL to Project Gutenberg replaced by _Georg Fischer_, Jan 03 2009

%E Edited by _M. F. Hasler_, Feb 24 2014

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Last modified April 20 01:36 EDT 2021. Contains 343117 sequences. (Running on oeis4.)