%I A006890 M3264
%S A006890 4,6,6,9,2,0,1,6,0,9,1,0,2,9,9,0,6,7,1,8,5,3,2,0,3,8,2,0,4,6,6,2,0,1,6,
%T A006890 1,7,2,5,8,1,8,5,5,7,7,4,7,5,7,6,8,6,3,2,7,4,5,6,5,1,3,4,3,0,0,4,1,3,4,
%U A006890 3,3,0,2,1,1,3,1,4,7,3,7,1,3,8,6,8,9,7,4,4,0,2,3,9,4,8,0,1,3,8,1,7,1,6
%N A006890 Decimal expansion of Feigenbaum bifurcation velocity.
%C A006890 "... These are related to properties of dynamical systems with 'period-doubling' 
               oscillations. The ratio of successive differences between period-doubling 
               bifurcation parameters approaches the number 4.669... Period doubling 
               has been discovered in many physical systems before they enter the 
               chaotic regime. Feigenbaum numbers have not been proved to be transcendental 
               but are generally believed to be. ..." [Pickover]
%C A006890 The Feigenbaum delta constant is the convergence ratio {g(k)-g(k-1)}/
               {g(k+1)-g(k)} of successive period-doubling thresholds g(k) in the 
               continuous map x(n+1)=f(x(n),g) of an interval onto itself. - Lekraj 
               Beedassy (blekraj(AT)yahoo.com), Jan 07 2005
%D A006890 K. Briggs, A precise calculation of the Feigenbaum constants, Math. Comp., 
               57 (1991), 435-439.
%D A006890 B. Derrida, A. Gervois and Y. Pomeau, Universal metric properties of 
               bifurcations, J. Phys. A 12 (1979), 269-.
%D A006890 S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and 
               its Applications, vol. 94, Cambridge University Press, pp. 65-76
%D A006890 C. A. Pickover, (1993) 'The fifteen most famous transcendental numbers.' 
               "Journal of Recreational Mathematics," 25(1):12.
%D A006890 C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind 
               and Meaning," Chapter 44, 'The 15 Most Famous Transcendental Numbers,
               ' Oxford University Press, Oxford, England, 2000, pages 103 - 106.
%D A006890 C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 462.
%D A006890 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A006890 Harry J. Smith, <a href="b006890.txt">Table of n, a(n) for n=1,...,1019</
               a>
%H A006890 C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind 
               and Meaning," <a href="http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?first=1&maxdocs=3&type=html&an\
               =0983.00008&format=complete">Zentralblatt review</a>
%H A006890 S. Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/
               math/MiscellaneousMathematicalConstants/chap33.html">Feigenbaum constants</
               a>
%H A006890 S. Plouffe, Plouffe's Inverter, <a href="http://pi.lacim.uqam.ca/piDATA/
               feigenbaum.txt">Feigenbaum constants to 1018 decimal places</a>
%H A006890 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               FeigenbaumConstant.html">Feigenbaum Constant</a>
%H A006890 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               FeigenbaumConstantApproximations.html">Feigenbaum Constant Approximations</
               a>
%H A006890 A. Krowne, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/
               FeigenbaumDeltaConstant.html">Feigenbaum constant</a>
%H A006890 Wikipedia, <a href="http://en.wikipedia.org/wiki/Feigenbaum_constant">
               Feigenbaum constant</a>
%H A006890 R. Munafo, <a href="http://www.mrob.com/pub/muency/feigenbaumconstant.html">
               Feigenbaum Constant</a>
%e A006890 4.669201609102990671853203820466201617258185577475768632745651343004134... 
               [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 13 2009]
%o A006890 Contribution from Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 15 
               2009: (Start)
%o A006890 (PARI) { default(realprecision,1019); delta=4.\
%o A006890 6692016091029906718532038204662016172581855774757686327456513430\
%o A006890 0413433021131473713868974402394801381716598485518981513440862714\
%o A006890 2027932522312442988890890859944935463236713411532481714219947455\
%o A006890 6443658237932020095610583305754586176522220703854106467494942849\
%o A006890 8145339172620056875566595233987560382563722564800409510712838906\
%o A006890 1184470277585428541980111344017500242858538249833571552205223608\
%o A006890 7250291678860362674527213399057131606875345083433934446103706309\
%o A006890 4520191158769724322735898389037949462572512890979489867683346116\
%o A006890 2688911656312347446057517953912204556247280709520219819909455858\
%o A006890 1946136877445617396074115614074243754435499204869180982648652368\
%o A006890 4387027996490173977934251347238087371362116018601281861020563818\
%o A006890 1835409759847796417390032893617143215987824078977661439139576403\
%o A006890 7760537119096932066998361984288981837003229412030210655743295550\
%o A006890 3888458497370347275321219257069584140746618419819610061296401614\
%o A006890 8771294441590140546794180019813325337859249336588307045999993837\
%o A006890 5411726563553016862529032210862320550634510679399023341675; x=delta; 
               for (n=1, 1019, d=floor(x); x=(x-d)*10; write("b006890.txt", n, " 
               ", d)); } (End)
%Y A006890 Cf. A006891, the Feigenbaum reduction parameter.
%Y A006890 Cf. A069544; A102817; A108952.
%Y A006890 Cf. A159766 = Continued fraction. [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), 
               May 13 2009]
%Y A006890 Sequence in context: A111653 A049089 A028327 this_sequence A104123 A094078 
               A016122
%Y A006890 Adjacent sequences: A006887 A006888 A006889 this_sequence A006891 A006892 
               A006893
%K A006890 cons,nonn,nice
%O A006890 1,1
%A A006890 N. J. A. Sloane (njas(AT)research.att.com), C. L. Mallows (colinm(AT)research.avayalabs.com), 
               Jeffrey Shallit
%E A006890 Additional comments from Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 29 
               2000
%E A006890 Fixed my PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), 
               May 19 2009