1,1

"... 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]

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

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

K. Briggs, A precise calculation of the Feigenbaum constants, Math. Comp., 57 (1991), 435-439.

B. Derrida, A. Gervois and Y. Pomeau, Universal metric properties of bifurcations, J. Phys. A 12 (1979), 269-.

C. A. Pickover, (1993) 'The fifteen most famous transcendental numbers.' "Journal of Recreational Mathematics," 25(1):12.

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.

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 65-76

C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 462.

Harry J. Smith, Table of n, a(n) for n=1,...,1019

C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review

S. Plouffe, Feigenbaum constants

S. Plouffe, Plouffe's Inverter, Feigenbaum constants to 1018 decimal places

Eric Weisstein's World of Mathematics, Feigenbaum Constant

Eric Weisstein's World of Mathematics, Feigenbaum Constant Approximations

A. Krowne, PlanetMath.org, Feigenbaum constant

Wikipedia, Feigenbaum constant

R. Munafo, Feigenbaum Constant

4.669201609102990671853203820466201617258185577475768632745651343004134... [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 13 2009]

Contribution from Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 15 2009: (Start)

(PARI) { default(realprecision, 1019); delta=4.\

6692016091029906718532038204662016172581855774757686327456513430\

0413433021131473713868974402394801381716598485518981513440862714\

2027932522312442988890890859944935463236713411532481714219947455\

6443658237932020095610583305754586176522220703854106467494942849\

8145339172620056875566595233987560382563722564800409510712838906\

1184470277585428541980111344017500242858538249833571552205223608\

7250291678860362674527213399057131606875345083433934446103706309\

4520191158769724322735898389037949462572512890979489867683346116\

2688911656312347446057517953912204556247280709520219819909455858\

1946136877445617396074115614074243754435499204869180982648652368\

4387027996490173977934251347238087371362116018601281861020563818\

1835409759847796417390032893617143215987824078977661439139576403\

7760537119096932066998361984288981837003229412030210655743295550\

3888458497370347275321219257069584140746618419819610061296401614\

8771294441590140546794180019813325337859249336588307045999993837\

5411726563553016862529032210862320550634510679399023341675; x=delta; for (n=1, 1019, d=floor(x); x=(x-d)*10; write("b006890.txt", n, " ", d)); } (End)

Cf. A006891, the Feigenbaum reduction parameter.

Cf. A069544; A102817; A108952.

Cf. A159766 = Continued fraction. [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 13 2009]

Adjacent sequences: A006887 A006888 A006889 this_sequence A006891 A006892 A006893

Sequence in context: A111653 A049089 A028327 this_sequence A104123 A094078 A016122

cons,nonn,nice

N. J. A. Sloane (njas(AT)research.att.com), C. L. Mallows (colinm(AT)research.avayalabs.com), Jeffrey Shallit

Additional comments from Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 29 2000

Fixed my PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 19 2009