A069544 -id:A069544 - cp's OEIS frontend

A069261 Denominators of the Egyptian fraction for the fractional part of Feigenbaum's constant, 4.6692...

2, 6, 395, 303319, 131209492876, 45596605913248081159007, 34243827483200809826686815883136413405197711755, 111445370519459209554489628949586784217535791333333948765270067675689059510906528783799426730444

Offset: 1

Christopher Lund (clund(AT)san.rr.com), Apr 14 2002

The next term in the series, a(9), is ~ 10^190.

The sequence gives the denominators for the fractional part of delta only. One could prefix four 1's in order to get (sum of reciprocals) = delta.

Kevin Ryde, Table of n, a(n) for n = 1..10

Cf. A006890 (Feigenbaum's constant), A069544 (continued fraction).

t=delta-4/*from A006890, or use: t=contfracpnqn(A069544); t[1,1]/t[2,1]*/; for(i=1,8,print1(1\t+1",");t-=1/(1\t+1)) \\ Requires delta to 93 decimals or A069544 to 90 terms (up to [...,1,1,4]) to get a(7) correctly, 180 terms for a(8). - M. F. Hasler, Apr 30 2018

a(n) = ceiling(1/(delta - 4 - Sum_{0 < i < n} 1/a(i))) is the smallest integer such that 4 + Sum_{i=1..n} 1/a(i) < delta = 4.6620... - M. F. Hasler, Apr 30 2018

Edited by M. F. Hasler, Apr 30 2018

A006890 Decimal expansion of Feigenbaum bifurcation velocity.

Original entry on oeis.org

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, 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, 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

Offset: 1

N. J. A. Sloane, Colin Mallows, Jeffrey Shallit

"... 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, Jan 07 2005
The above statement is only valid for functions f satisfying some properties, e.g., having a single locally quadratic maximum. See, e.g., the MathWorld link for more details. - M. F. Hasler, May 01 2018
Named after the American mathematical physicist Mitchell Jay Feigenbaum (1944-2019). - Amiram Eldar, Jun 16 2021

			4.669201609102990671853203820466201617258185577475768632745651343004134...

Michael F. Barnsley, Fractals Everywhere, New Edition, Prof. of Math., Australian National University, Dover Publications, Inc., Mineola, NY, 2012, page 314.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 208.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 65-76
Clifford A. Pickover, (1993) 'The fifteen most famous transcendental numbers.' "Journal of Recreational Mathematics," 25(1):12.
Clifford 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.
Clifford A. Pickover, The Math Book, Sterling, NY, 2009; see p. 462.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Ian Stewart, Nature's Numbers, Chapter 8, Do Dice Play God?, Weidenfeld & Nicolson, 1995.

Harry J. Smith, Table of n, a(n) for n = 1..1019
Keith Briggs, A precise calculation of the Feigenbaum constants, Math. Comp., Vol. 57, No. 195 (1991), pp. 435-439.
B. Derrida, A. Gervois and Y. Pomeau, Universal metric properties of bifurcations, J. Phys. A, Vol. 12 (1979), pp. 269-296.
Brady Haran and Phillip Moriarty, A magic number (video) (2009).
Brady Haran and Ben Sparks, 4.669, Numberphile video (2017).
Sibyl Kempson, Restless Eye: Text for the Advanced Beginner Group, PAJ: A Journal of Performance and Art, Volume 34, Number 3, September 2012 (PAJ 102).
A. Krowne, Feigenbaum constant, PlanetMath.org.
Robert P. Munafo, Feigenbaum Constant.
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review.
Simon Plouffe, Feigenbaum constants.
Simon Plouffe, Feigenbaum constants to 1018 decimal places.
Gianluca Simonetto, Chaos and universality in non-linear dynamics: the logistic map, Univ. Padova (Italy, 2023, in Italian).
Judi Thurlby, Rigorous calculations of renormalisation fixed points and attractors, PhD thesis, U. Portsmouth, (2021). 400 digits in section 3.8.
Eric Weisstein's World of Mathematics, Feigenbaum Constant.
Eric Weisstein's World of Mathematics, Feigenbaum Constant Approximations.
Wikipedia, Feigenbaum constant.

Cf. A159766 and A069544 (continued fraction), A069261 (Egyptian fraction), A108952 (1/delta), A102817 (Gamma(delta^2)).

Cf. A006891 (Feigenbaum reduction parameter), A218453.

Additional comments from Robert G. Wilson v, Dec 29 2000

A159766 Continued fraction expansion of the Feigenbaum constant A006890 = 4.66920160910...

Original entry on oeis.org

4, 1, 2, 43, 2, 163, 2, 3, 1, 1, 2, 5, 1, 2, 3, 80, 2, 5, 2, 1, 1, 1, 33, 1, 1, 53, 1, 1, 1, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 239, 1, 3, 31, 1, 1, 11, 1, 13, 123, 2, 2, 2, 2, 13, 15, 1, 2, 3, 3, 1, 3, 1, 1, 6, 1, 3, 1, 1, 13, 8, 1, 7, 1, 2, 1, 8, 7, 1, 17, 1, 6, 1, 1, 3, 1, 1, 13, 1, 1, 4, 2, 9, 124, 1, 1, 3

Offset: 0

Harry J. Smith, Apr 21 2009

Feigenbaum bifurcation velocity delta. See A006890 for further information.

Apart from the first term, the same as A069544. - R. J. Mathar, Apr 28 2009

			4.66920160910299067185320382... = 4 + 1/(1 + 1/(2 + 1/(43 + 1/(2 + ...))))

Harry J. Smith, Table of n, a(n) for n = 0..964 [shortened by _Kevin Ryde_, Sep 30 2024]

Cf. A006890 (decimal expansion of the constant), A069544 (continued fraction of the fractional part), A069261 (Egyptian fraction).

fbvd = (* copy value from the link in A006890 or immediately above *); ContinuedFraction@ fbvd (* Robert G. Wilson v, Apr 27 2009 *)

{ default(realprecision,1019); delta=4.\
6692016091029906718532038204662016172581855774757686327456513430\
0413433021131473713868974402394801381716598485518981513440862714\
2027932522312442988890890859944935463236713411532481714219947455\
6443658237932020095610583305754586176522220703854106467494942849\
8145339172620056875566595233987560382563722564800409510712838906\
1184470277585428541980111344017500242858538249833571552205223608\
7250291678860362674527213399057131606875345083433934446103706309\
4520191158769724322735898389037949462572512890979489867683346116\
2688911656312347446057517953912204556247280709520219819909455858\
1946136877445617396074115614074243754435499204869180982648652368\
4387027996490173977934251347238087371362116018601281861020563818\
1835409759847796417390032893617143215987824078977661439139576403\
7760537119096932066998361984288981837003229412030210655743295550\
3888458497370347275321219257069584140746618419819610061296401614\
8771294441590140546794180019813325337859249336588307045999993837\
5411726563553016862529032210862320550634510679399023341675; A159766=contfrac(delta); for (n=1, 967, write("b159766.txt", n-1, " ", A159766[n])); } \\ Harry J. Smith, May 15 2009

A159766=contfrac(delta) \\ M. F. Hasler, Apr 30 2018

Old PARI program deleted by Harry J. Smith, May 19 2009

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A069261 Denominators of the Egyptian fraction for the fractional part of Feigenbaum's constant, 4.6692...

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A006890 Decimal expansion of Feigenbaum bifurcation velocity.

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A159766 Continued fraction expansion of the Feigenbaum constant A006890 = 4.66920160910...

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