A086178 -id:A086178 - cp's OEIS frontend

A090488 Decimal expansion of 2 + 2*sqrt(2).

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

Offset: 1

Felix Tubiana, Feb 05 2004

Side length of smallest square containing five circles of radius 1. - Charles R Greathouse IV, Apr 05 2011
Equals n + n/(n +n/(n +n/(n +....))) for n = 4.  See also A090388. - Stanislav Sykora, Jan 23 2014
Also the area of a regular octagon with unit edge length. - Stanislav Sykora, Apr 12 2015
The positive solution to x^2 - 4*x - 4 = 0. The negative solution is -1 * A163960 = -0.82842... . - Michal Paulovic, Dec 12 2023

			4.828427124746190097603377448419396157139343750...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Octagon
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 2

Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090654 (n=8), A090655 (n=9), A090656 (n=10). - Stanislav Sykora, Jan 23 2014
Cf. A002193, A010466, A014176, A086178, A156035, A157258, A163960, A365823.
Cf. Areas of other regular polygons: A120011, A102771, A104956, A178817, A256853, A178816, A256854, A178809.

RealDigits[2+2Sqrt[2],10,120][[1]] (* Harvey P. Dale, Mar 11 2015 *)

2*(1 + sqrt(2)) \\ G. C. Greubel, Jul 03 2017

Equals 1 + A086178 = 2*A014176. - R. J. Mathar, Sep 03 2007
From Michal Paulovic, Dec 12 2023: (Start)
Equals A010466 + 2.
Equals A156035 - 1.
Equals A157258 - 5.
Equals A163960 + 4.
Equals A365823 - 2.
Equals [4; 1, 4, ...] (periodic continued fraction expansion).
Equals sqrt(4 + 4 * sqrt(4 + 4 * sqrt(4 + 4 * sqrt(4 + 4 * ...)))). (End)

Better definition from Rick L. Shepherd, Jul 02 2004

A086180 Decimal expansion of 1 + sqrt(6).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jul 11 2003

Onset of 4-cycle in the logistic equation.

			3.4494897427831780981972840747058913919659474806567...

Gian Italo Bischi, Rosa Carini, Laura Gardini, and Paolo Tenti, Sulle orme del caos: comportamenti complessi in modelli matematici semplici. Bruno Mondadori (Milano), 2004. See pp. 83, 92, 95. (In Italian)
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.9, p. 66.

Andy Burbanks, Period Doubling: Iteration of the Logistic Map.
Kurt Cogswell, WebDiagram Applet.
Michael Cross, The Period Doubling Route To Chaos.
Blair Fraser, Iterated Maps.
Alan Hood, Logistic Equation.
Pete Kernan, Logistic Map(Cobweb Construction).
Pete Kernan, Logistic Map(Time Series).
Oliver Knill, Attractor of quadratic map.
Daniel Marthaler, Logistic Map.
John McDonald, Iteration versus time for the Logistic map.
Frank Wattenberg, An Applet in a Window--Cobweb Diagrams.
Eric Weisstein's World of Mathematics, Logistic Map.
Index entries for algebraic numbers, degree 2.

Cf. A010464, A086178.

RealDigits[1+Sqrt[6],10,120][[1]] (* Harvey P. Dale, Jul 28 2012 *)

1+sqrt(6) \\ Stefano Spezia, Nov 17 2024

Equals A010464 plus 1. - R. J. Mathar, Sep 12 2008

A086179 Decimal expansion of upper bound for the r-values supporting stable period-3 orbits in the logistic equation.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jul 11 2003

			3.84149900754...

Eric Weisstein's World of Mathematics, Logistic Map
Index entries for algebraic numbers, degree 6.

Cf. A086178.

RealDigits[Root[ -81-36*#1-14*#1^2+24*#1^3+4*#1^4-6*#1^5+#1^6&, 2], 10, 110][[1]]

polrootsreal(x^6-6*x^5+4*x^4+24*x^3-14*x^2-36*x-81)[2] \\ Charles R Greathouse IV, Feb 03 2025

A118452 Decimal expansion of onset of logistic map 5-bifurcation.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Apr 28 2006

Algebraic of order 22.

			3.7381723752659623694...

Eric Weisstein's World of Mathematics, Logistic Map

Cf. A086178, A086180, A091517, A098587, A118453, A118454.

RealDigits[Root[ -28629151 - 31399714*#1 - 14329471*#1^2 + 3613360*#1^3 + 13369804*#1^4 + 5836984*#1^5 - 1495060*#1^6 - 4002880*#1^7 - 1613676*#1^8 + 1484268*#1^9 + 808298*#1^10 - 234632*#1^11 - 294146*#1^12 + 8008*#1^13 + 103556*#1^14 - 17344*#1^15 - 18092*#1^16 + 7488*#1^17 + 116*#1^18 - 700*#1^19 + 189*#1^20 - 22*#1^21 + #1^22 &, 4],10,110][[1]]

A181906 The value of r at the onset of the first period-9 cycle of the logistic map f(x) = rx(1 - x).

Original entry on oeis.org

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

Offset: 1

Cheng Zhang, Apr 01 2012

Root of a degree 240*2 = 480 polynomial.

			3.6871968733...

Cheng Zhang, the polynomial for T = r*(r-2)

Cf. A086178, A086179, A086180, A086181, A091517, A118452, A118453, A118746, A181907, A181909, A181910, A181911, A181912, A181913, A181915, A181916, A181917, A181918, A181919.

A181907 The value of r at the onset of the first period-10 cycle of the logistic map f(x) = rx(1 - x).

Original entry on oeis.org

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

Offset: 1

Cheng Zhang, Apr 01 2012

Root of a degree 472*2 = 944 polynomial.

			3.60520806689...

Cheng Zhang, the polynomial for T = r*(r-2)

Cf. A086178, A086179, A086180, A086181, A091517, A118452, A181906, A181909, A181910, A181911, A181912, A181913, A181915, A181916, A181917, A181918, A181919.

A181909 The r value at the onset of the first period-11 cycle of the logistic map f(x) = rx(1 - x).

Original entry on oeis.org

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

Offset: 1

Cheng Zhang, Apr 01 2012

Root of a degree 1013*2 = 2026 polynomial.

			3.6817160193...

Cheng Zhang, the polynomial for T = r*(r-2)

Cf. A086178, A086179, A086180, A086181, A091517, A118452, A118453, A118746, A181906, A181907, A181910, A181911, A181912, A181913, A181915, A181916, A181917, A181918, A181919.

A181910 The value of r at the onset of the first period-12 cycle of the logistic map f(x) = rx(1 - x).

Original entry on oeis.org

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

Offset: 1

Cheng Zhang, Apr 01 2012

Root of a degree 1959*2 = 3918 polynomial.

			3.58202300107...

Cheng Zhang, the polynomial for T = r*(r-2)

Cf. A086178, A086179, A086180, A086181, A091517, A118452, A118453, A118746, A181906, A181907, A181909, A181911, A181912, A181913, A181915, A181916, A181917, A181918, A181919

A181911 The value of r at the onset of the first period-13 cycle of the logistic map f(x) = rx(1 - x).

Original entry on oeis.org

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

Offset: 1

Cheng Zhang, Apr 01 2012

Root of a degree 4083*2 = 9166 polynomial.

			3.6797024577659...

Cheng Zhang, the polynomial for T = r*(r-2)

Cf. A086178, A086179, A086180, A086181, A091517, A118452, A118453, A118746, A181906, A181907, A181909, A181910, A181912, A181913, A181915, A181916, A181917, A181918, A181919

A118453 Decimal expansion of onset of logistic map 6-bifurcation.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Apr 28 2006

Algebraic of order 40.

			3.6265531616949737258...

Eric Weisstein's World of Mathematics, Logistic Map

Cf. A086178, A086180, A091517, A098587, A118454.

RealDigits[Root[3063651608241 + 583552687284*#1 + 1847916843066*#1^2 - 195203691396*#1^3 - 266965430067*#1^4 - 930539982708*#1^5 - 32353949026*#1^6 - 20846657216*#1^7 + 268499651644*#1^8 + 203273817100*#1^9 - 70347416426*#1^10 - 23477492116*#1^11 - 148805533357*#1^12 + 63425976316*#1^13 + 29299283358*#1^14 + 1219753488*#1^15 + 10442067223*#1^16 - 25709485160*#1^17 + 5757007476*#1^18 + 945192256*#1^19 + 3195434424*#1^20 + 746246488*#1^21 - 3839476860*#1^22 + 1483852216*#1^23 + 347099271*#1^24 - 126361652*#1^25 - 100717898*#1^26 - 75365092*#1^27 + 137026587*#1^28 - 48885412*#1^29 - 12966642*#1^30 + 16770304*#1^31 - 5563348*#1^32 + 195228*#1^33 + 516582*#1^34 - 225060*#1^35 + 52181*#1^36 - 7676*#1^37 + 722*#1^38 - 40*#1^39 + #1^40 &, 5],10,110][[1]]

cp's OEIS Frontend

A090488 Decimal expansion of 2 + 2*sqrt(2).

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A086180 Decimal expansion of 1 + sqrt(6).

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A086179 Decimal expansion of upper bound for the r-values supporting stable period-3 orbits in the logistic equation.

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A118452 Decimal expansion of onset of logistic map 5-bifurcation.

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A181906 The value of r at the onset of the first period-9 cycle of the logistic map f(x) = r*x*(1 - x).

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A181907 The value of r at the onset of the first period-10 cycle of the logistic map f(x) = r*x*(1 - x).

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A181909 The r value at the onset of the first period-11 cycle of the logistic map f(x) = r*x*(1 - x).

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A181910 The value of r at the onset of the first period-12 cycle of the logistic map f(x) = r*x*(1 - x).

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A181911 The value of r at the onset of the first period-13 cycle of the logistic map f(x) = r*x*(1 - x).

A181906 The value of r at the onset of the first period-9 cycle of the logistic map f(x) = rx(1 - x).

A181907 The value of r at the onset of the first period-10 cycle of the logistic map f(x) = rx(1 - x).

A181909 The r value at the onset of the first period-11 cycle of the logistic map f(x) = rx(1 - x).

A181910 The value of r at the onset of the first period-12 cycle of the logistic map f(x) = rx(1 - x).

A181911 The value of r at the onset of the first period-13 cycle of the logistic map f(x) = rx(1 - x).