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%I A001622 M4046 N1679
%S A001622 1,6,1,8,0,3,3,9,8,8,7,4,9,8,9,4,8,4,8,2,0,4,5,8,6,8,3,4,3,6,5,6,3,8,1,
%T A001622 1,7,7,2,0,3,0,9,1,7,9,8,0,5,7,6,2,8,6,2,1,3,5,4,4,8,6,2,2,7,0,5,2,6,0,
%U A001622 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 A001622 Decimal expansion of golden ratio phi (or tau) = (1 + sqrt 5 )/2.
%C A001622 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
%C A001622 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
%C A001622 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]
%D A001622 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001622 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001622 M. Berg, Phi, the golden ratio (to 4599 decimal places) and Fibonacci 
               numbers, Fib. Quart., 4 (1961), 157-162.
%D A001622 R. A. Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific, 
               River Edge NJ 1997.
%D A001622 S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and 
               its Applications, vol. 94, Cambridge University Press, Section 1.2.
%D A001622 M. Gardner, The Second Scientific American Book Of Mathematical Puzzles 
               and Diversions, "Phi:The Golden Ratio", Chapter 8, Simon & Schuster 
               NY 1961.
%D A001622 H. E. Huntley, The Divine Proportion, Dover NY 1970.
%D A001622 M. Livio, The Golden Ratio, Broadway Books, NY, 2002.
%D A001622 S. Olsen, The Golden Section, Walker & Co. NY 2006.
%D A001622 H. Walser, The Golden Section, Math. Assoc. of Amer. Washington DC 2001.
%D A001622 C. J. Willard, Le nombre d'or, Magnard Paris 1987.
%D A001622 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]
%H A001622 Robert G. Wilson v, <a href="b001622.txt">Table of n, a(n) for n=1..100000</
               a>
%H A001622 John Baez, <a href="http://math.ucr.edu/home/baez/week203.html">This 
               week's finds in mathematical physics, Week 203</a>
%H A001622 A. Camus College Team, <a href="http://www.col-camus-soufflenheim.ac-strasbourg.fr/
               Page.php?IDP=135">Le nombre d'or</a>
%H A001622 T. Eveilleau, <a href="http://perso.orange.fr/therese.eveilleau/pages/
               truc_mat/textes/rectangle_dor.htm">Le nombre d'or(Text in French)</
               a>
%H A001622 Gutenberg Project, <a href="http://www.gutenberg.org/etext/633">The golden 
               ratio to 20000 places</a>
%H A001622 Heartbeat200.com, <a href="http://www.heartbeat2000.com/phi.htm">Introduction 
               to The Golden Proportion</a>
%H A001622 ICON Project, <a href="http://www.cs.arizona.edu/icon/oddsends/phi.htm">
               The golden ratio to 50000 places</a>
%H A001622 R. Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/
               ">Fibonacci numbers and the golden section</a>
%H A001622 E. Levin, <a href="http://www.goldenmeangauge.co.uk/golden.htm">The Golden 
               Proportion</a>
%H A001622 Mathematical Database, Poster, <a href="http://mathdb.org/gallery/poster/
               description/e_poster_04.htm">The Golden Ratio</a>
%H A001622 Meiner, <a href="http://goldennumber.net">Phi:The Golden Number</a>
%H A001622 D. Merrill, <a href="http://home.netcom.com/~merrills">Fib-Phi Link Page</
               a>
%H A001622 D. Merrill, <a href="http://home.netcom.com/~merrills/phi1000000.html">
               Golden ratio to 1000000 digits</a>
%H A001622 J. C. Michel, <a href="http://jc.michel.free.fr/nombre_d_or.php">Le nombre 
               d'or</a>
%H A001622 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 A001622 S. Plouffe, Plouffe's Inverter, <a href="http://pi.lacim.uqam.ca/piDATA/
               golden.txt">The golden ratio to 10 million digits</a>
%H A001622 S. Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/
               math/MiscellaneousMathematicalConstants/chap46.html">The golden ratio:(1+sqrt(5))/
               2 to 20000 places</a>
%H A001622 F. Richman, Fibonacci sequence with multiprecision Java, <a href="http:/
               /www.math.fau.edu/Richman/fibjava.htm">Successive approximations 
               to phi from ratios of consecutive Fibonacci numbers </a>
%H A001622 E. F. Schubert, <a href="http://www.rpi.edu/~schubert/Educational%20resources/
               Fibonacci%20series.pdf">The Fibonacci series</a>
%H A001622 A. M. Selvam, <a href="http://members.tripod.com/~amselvam/cycas/cycas.html">
               Golden mean and self-similar,fractal geometrical structures in nature</
               a>
%H A001622 M. R. Watkins, <a href="http://www.maths.ex.ac.uk/~mwatkins/zeta/goldenmean.htm">
               The "Golden Mean" in number theory</a>
%H A001622 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               GoldenRatio.html">Link to a section of The World of Mathematics.</
               a>
%H A001622 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SilverRatio.html">Silver Ratio</a>
%H A001622 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>
%H A001622 Wikipedia, <a href="http://www.wikipedia.org/wiki/Golden_mean">Golden 
               mean</a>
%H A001622 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. [From William Rex Marshall (w.r.marshall(AT)actrix.co.nz), 
               Aug 27 2008]
%H A001622 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. [From William Rex Marshall (w.r.marshall(AT)actrix.co.nz), 
               Aug 27 2008]
%F A001622 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).
%F A001622 General formula: Provided x>1 suffices x-x^(-1)=floor(x), where x=phi 
               for this sequence, then
%F A001622 for odd n: x^n-x^(-n)=floor(x^n), hence fract(x^n)=x^(-n),
%F A001622 for even n: x^n+x^(-n)=ceiling(x^n), hence fract(x^n)=1-x^(-n),
%F A001622 for all n>0: x^n+(-x)^(-n)=nint(x^n).
%F A001622 x=phi is the minmal solution to x-x^(-1)=floor(x) (where floor(x)=1 in 
               this case).
%F A001622 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)
%e A001622 1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391138...
%t A001622 RealDigits[(1 + Sqrt[5])/2, 10, 130] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 02 2006
%t A001622 RealDigits[ Exp[ ArcSinh[1/2]], 10, 111][[1]] (* Robert G. Wilson v (rgwv(AT)rgwv.com), 
               Mar 01 2008 *)
%o A001622 (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]
%o A001622 Contribution from Michael Porter (michael_b_porter(AT)yahoo.com), Oct 
               24 2009: (Start)
%o A001622 (PARI) /* Digit-by-digit method */
%o A001622 /* write it as 0.5+sqrt(1.25) and start at hundredths digit */
%o A001622 r=11; x=400; print(1); print(6);
%o A001622 for(digits=1, 110, {d=0; while((20*r+d)*d <= x, d++);
%o A001622 d--; /* while loop overshoots correct digit */
%o A001622 print(d); x=100*(x-(20*r+d)*d); r=10*r+d}) (End)
%Y A001622 Cf. A000012.
%Y A001622 Cf. A104457.
%Y A001622 A145996 [From Artur Jasinski (grafix(AT)csl.pl), Oct 26 2008]
%Y A001622 Cf. A000032, A006497, A080039.
%Y A001622 Sequence in context: A143019 A156921 A094214 this_sequence A021622 A073228 
               A145314
%Y A001622 Adjacent sequences: A001619 A001620 A001621 this_sequence A001623 A001624 
               A001625
%K A001622 cons,nonn,nice,easy
%O A001622 1,2
%A A001622 N. J. A. Sloane (njas(AT)research.att.com).
%E A001622 Additional links contributed by Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Dec 23 2003
%E A001622 More terms from Gabriel Cunningham (gcasey(AT)mit.edu), Oct 24 2004
%E A001622 More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 02 2006
%E A001622 Broken URL to Project Gutenberg replaced by Dr. Georg Fischer (Georg.Fischer(AT)T-Online.de), 
               Jan 03 2009
%E A001622 Corrected PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), 
               May 17 2009

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