A006891 -id:A006891 - cp's OEIS frontend

A159767 Continued fraction expansion of A006891.

2, 1, 1, 85, 2, 8, 1, 10, 16, 3, 8, 9, 2, 1, 40, 1, 2, 3, 2, 2, 1, 17, 1, 1, 5, 3, 2, 6, 3, 5, 1, 1, 3, 3, 15, 3, 1, 1, 7, 2, 3, 1, 7, 2, 1, 55, 1, 1, 1, 1, 4, 9, 1, 2, 1, 36, 1, 5, 10, 1, 1, 2, 1, 4, 1, 4, 5, 5, 1, 1, 130, 1, 3, 1, 1, 2, 1, 3, 1, 3, 2, 3, 2, 2, 547, 9, 18, 3, 1, 4, 2, 1, 1, 2, 2, 2, 1, 6, 1

Offset: 0

Harry J. Smith, Apr 21 2009

Feigenbaum bifurcation velocity alpha.

			-alpha = 2.50290787509589282... = 2 + 1/(1 + 1/(1 + 1/(85 + 1/(2 + ...))))

Harry J. Smith, Table of n, a(n) for n = 0..995

(* assign to 'x' the Feigenbaum bifurcation velocity alpha *); ContinuedFraction[ x, 100] (* Robert G. Wilson v, May 31 2009 *)

{ default(realprecision,1019); alpha=-2.\
5029078750958928222839028732182157863812713767271499773361920567\
7923546317959020670329964974643383412959523186999585472394218237\
7785445179272863314993372578112163594879503744781260997380598671\
2397117373289276654044010306698313834600094139322364490657889951\
2205843172507873377463087853424285351988587500042358246918740820\
4281700901714823051821621619413199856066129382742649709844084470\
1008054549677936760888126446406885181552709324007542506497157047\
0475419932831783645332562415378693957125097066387979492654623137\
6745918909813116752434221110130913127837160951158341230841503716\
4997020224681219644081216686527458043026245782561067150138521821\
6449532543349873487413352795815351016583605455763513276501810781\
1948369459574850237398235452625632779475397269902012891516645793\
9420198920248803394051699686551494477396533876979741232354061781\
9896112494095990353128997733611849847377946108428833293833903950\
9008914086351525626803381414669279913310743349705143545201344643\
4264752001621384610729922641994332772918977769053802596851; x=contfrac(-alpha); for (n=1, 996, write("b159767.txt", n-1, " ", x[n])); } (End)

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

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

A103546 Decimal expansion of the negated value of the smallest real root of the quintic equation x^5 + 2x^4 - 2x^3 - x^2 + 2*x -1 = 0.

Original entry on oeis.org

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

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Mar 23 2005

This is an approximation to the Feigenbaum reduction parameter.

The other two real roots are 0.76660865407289... and -1.16317291980104...

			The real roots are (roughly) -2.486343765, -1.163172920, 0.7666086541.

Eric Weisstein's World of Mathematics, Feigenbaum Constant.
Index entries for algebraic numbers, degree 5

Cf. A006891, A103616.

RealDigits[ FindRoot[x^5 + 2x^4 - 2x^3 - x^2 + 2x - 1 == 0, {x, -3}, WorkingPrecision -> 2^7][[1, 2]]][[1]] (* Robert G. Wilson v, Mar 26 2005 *)
Root[#^5 + 2#^4 - 2#^3 - #^2 + 2# - 1&, 1] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Feb 27 2013 *)

polrootsreal(x^5 - 2*x^4 - 2*x^3 + x^2 + 2*x + 1)[3] \\ Charles R Greathouse IV, Apr 14 2014

A102817 Decimal expansion of Gamma(delta)^2 where delta is the Feigenbaum bifurcation velocity constant (A006890).

Original entry on oeis.org

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

Offset: 3

Gerald McGarvey, Feb 26 2005

Let x be this constant, then Integral_{t=1..x} sin(t)/sqrt(t) dt = 0.655555692248871113068...
delta^2 = 21.8014436664499573..., (delta/Gamma(delta))^2 = 0.10000663312663433933000349...
If s is solution of Gamma(s) - sqrt(218) = 0 then 1/((s - delta)*Gamma(delta)^6) = 2.5555951358396... whereas a^(Pi/4) = 2.055596478435... where a is Feigenbaum alpha constant (A006891), the difference = 0.4999986574... ~ 1/(2 + 10^-5.27)
10*cos(Gamma(delta)^2) + Pi = -0.199999019922688714710053...

			217.99997644999881468628813957793609890726797890973...

D. Broadhurst, Feigenbaum constants to 1018 decimal places

Cf. A006890, A006891.

Set delta then RealDigits[Gamma[delta]^2, 10, 110][[1]]

acos(Pi/10+.0199999019922688714710053)+69*Pi \\ Yields ~ 30 digits. Using (2e5-1)/(1e7-1) yields ~ 15 digits. For a better value use, e.g., delta from the Broadhurst link. - M. F. Hasler, Apr 30 2018

A218453 Decimal expansion of the Myrberg point.

Original entry on oeis.org

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

Offset: 1

Michael Barnsley (Michael.Barnsley(AT)anu.edu.au) and Robert G. Wilson v, Oct 28 2012

"3.3 The sequence of bubbles {B_n}_{n=0...oo} in exercise 3.2 converges to the 'Myreberg point', lambda = 1.40115...." [Barnsley, 2012, p.314]

[Author M. Barnsley consistently spells incorrectly "Myreberg" the name of Pekka J. Myrberg (1892-1976). We leave the original spelling in this citation to improve search results for this phrase. - M. F. Hasler, May 01 2018]

			1.4011551890920506005238267878938612922263...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.8.3, p. 439.

Michael F. Barnsley, Fractals Everywhere, New Edition, Dover Publications, Inc., Mineola, NY, 2012, page 314. ISBN:9780486488707.
Michael F. Barnsley, SuperFractals, Patterns of Nature, Cambridge University Press, United Kingdom, 2006. ISBN 9780521844932.
Michael F. Barnsley, Superfractals (personal web site).
David E. Joyce, Julia and Mandelbrot Sets, Clark University, 1994.
Faber McMullen, The Edge of Chaos (personal web page, updated April 2018).
Wikipedia, Pekka Myrberg.

Cf. A006890, A006891.

More terms from Joerg Arndt, Oct 31 2012
Removed the terms given Oct 31 2012 (these corresponded to the fixed point of f^(2^24) on the real axis). - Joerg Arndt, Jul 17 2016
More terms from Michael Lontke, Irena Rubtsova, and Joerg Arndt, Jan 16 2017

A101419 Decimal expansion of d - log(d-e) where d is Feigenbaum's bifurcation velocity constant (A006890, delta constant) and e is Euler's constant.

Original entry on oeis.org

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

Offset: 1

Gerald McGarvey, Jan 16 2005

From Plouffe's Inverter: log(s)^2 * a^2 * sin(1)^1 = 4.000900694855076... where s is Sierpinski's constant (A062089) and a is Feigenbaum's reduction parameter (A006891, alpha constant). 20 + Pi - exp(Pi) = .00090002081...

			4.00090066535299100480962602422024957759530322082790880094053997689...

Cf. A006890, A006891, A001113, A062089.

A195102 Decimal expansion of the reduction parameter for the biquadratic solution of the Feigenbaum-Cvitanovic equation.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Sep 09 2011

The expansion which starts quadratically 1+O(z^2) near the origin is in A006891. The constant here refers to the solution which starts 1+O(z^4) near z=0.

			Equals 1.690302971405244853...

K. M. Briggs, G. R. W. Quispel, C. J. Thompson, Feigenvalues for Mandelsets, J. Phys. A 24 (1991), 3363-3368.
R. J. Mathar, Chebyshev series representation of Feigenbaum's period-doubling function, arXiv:1008.4608 [math.DS], 2010, Section 3.3.
J. Thurlby, Rigorous calculations of renormalisation fixed points and attractors, PhD thesis, U. Portsmouth, (2021). Gives approx 330 digits on page 86.

Cf. A006891.

A306963 Decimal expansion of Feigenbaum's constant 0.399535...

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Mar 18 2019

			0.3995352805231344898575...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.9, pp. 66-67.

M. Campanino, H. Epstein, and D. Ruelle, On Feigenbaum's functional equation g o g (lambda x) + lambda g(x) = 0, Topology, Vol. 21, No. 2 (1982), pp. 125-129.
Artem Dudko and Scott Sutherland, On the Lebesgue measure of the Feigenbaum Julia set, Inventiones mathematicae, Vol. 221 (2020), pp. 167-202.
Mitchell J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys., Vol. 19, No. 1 (1978), pp. 25-52; alternative link; Wayback Machine copy.
Mitchell J. Feigenbaum, The universal metric properties of nonlinear transformations, J. Statist. Phys., Vol. 21, No. 6 (1979), pp. 669-706; CiteSeerX; Wayback Machine copy.
J. Thurlby, Rigorous calculations of renormalisation fixed points and attractors, PhD thesis, U. Portsmouth, (2021). 400 digits in section 3.8.

Cf. A006890, A006891, A119277, etc.

Equals 1/A006891. - Stefano Spezia, Nov 23 2024

More terms from Dudko and Sutherland (2020) added by Amiram Eldar, May 15 2021

a(22)-a(99) from Stefano Spezia, Nov 23 2024

cp's OEIS Frontend

A159767 Continued fraction expansion of A006891.

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A006890 Decimal expansion of Feigenbaum bifurcation velocity.

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A103546 Decimal expansion of the negated value of the smallest real root of the quintic equation x^5 + 2*x^4 - 2*x^3 - x^2 + 2*x -1 = 0.

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A102817 Decimal expansion of Gamma(delta)^2 where delta is the Feigenbaum bifurcation velocity constant (A006890).

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A218453 Decimal expansion of the Myrberg point.

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A101419 Decimal expansion of d - log(d-e) where d is Feigenbaum's bifurcation velocity constant (A006890, delta constant) and e is Euler's constant.

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A195102 Decimal expansion of the reduction parameter for the biquadratic solution of the Feigenbaum-Cvitanovic equation.

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A306963 Decimal expansion of Feigenbaum's constant 0.399535...

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A103546 Decimal expansion of the negated value of the smallest real root of the quintic equation x^5 + 2x^4 - 2x^3 - x^2 + 2*x -1 = 0.