A086180 -id:A086180 - cp's OEIS frontend

A181918 The value of r at the bifurcation point of the first period-12 cycle of the logistic map f(x) = rx(1 - x).

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

Offset: 1

Cheng Zhang, Apr 01 2012

Root of a degree 2010*2 = 4020 polynomial

			3.5828117795...

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

Cf. A086178, A086179, A086180, A086181, A091517, A118452, A181906, A181907, A181909, A181910, A181911, A181912, A181913, A181915, A181916, A181917, A181919.

A181919 The value of r at the bifurcation point 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, 3, 8, 4, 9, 8, 0, 3, 2, 9, 4, 7, 3, 0, 2, 7, 1, 7, 2, 8, 9, 8, 8, 1, 5, 7, 7, 3, 5, 7, 8, 2, 1, 1, 6, 7, 5, 6, 9, 1, 5, 0, 3, 3, 2, 5, 1, 5, 9, 3, 9, 6, 9, 6, 3, 4, 9, 5, 7, 8, 3, 3, 0, 7, 5, 2, 8, 5, 7, 4, 5, 0, 9, 8, 2, 6, 2, 4, 8, 2, 1, 0, 6, 9, 0, 5, 1, 7, 2, 2, 2, 4, 0, 3, 8

Offset: 1

Cheng Zhang, Apr 01 2012

Root of a degree 4095*2 = 8190 polynomial.

			3.6797038498...

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

Cf. A086178, A086179, A086180, A086181, A091517, A118452, A181906, A181907, A181909, A181910, A181911, A181912, A181913, A181915, A181916, A181917, A181918.

A181912 The value of r at the bifurcation point of the first period-5 cycle of the logistic map f(x) = rx(1 - x).

Original entry on oeis.org

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

Offset: 1

Cheng Zhang, Apr 01 2012

Root of a degree 15*2 = 30 polynomial.

			3.7411207566...

Cf. A086178, A086179, A086180, A086181, A091517, A118452, A118453, A118746, A181906, A181907, A181909, A181910, A181911, A181913, A181915, A181916, A181917, A181918, A181919.

RealDigits[1 + Sqrt[1 + T] /. NSolve[1291467969 - 313083144 T + 149426046 T^2 - 88548768 T^3 + 58697100 T^4 - 26978787 T^5 + 11351480 T^6 - 4444924 T^7 + 1519712 T^8 - 462764 T^9 + 118147 T^10 - 24008 T^11 + 3838 T^12 - 448 T^13 + 32 T^14 - T^15 == 0, T, Reals, WorkingPrecision -> 200][[1]][[1]]][[1]]

A181913 The value of r at the bifurcation point of the first period-7 cycle of the logistic map f(x) = rx(1 - x).

Original entry on oeis.org

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

Offset: 1

Cheng Zhang, Apr 01 2012

Root of a degree 63*2 = 126 polynomial.

			3.702154928...

Cf. A086178, A086179, A086180, A086181, A091517, A118452, A118453, A118746, A181906, A181907, A181909, A181910, A181911, A181912, A181915, A181916, A181917, A181918, A181919.

RealDigits[1 + Sqrt[1 + T] /.  NSolve[97862157334118736160267353892330031361 - 24275883989858911295570196314376441888 T + 11949756847721247033090755550100031472 T^2 - 7305759525507048491687489710934851842 T^3 + 4979912078948645588349153608449721856 T^4 - 3626559126667087845228068253830569728 T^5 + 2762422187660818660072532819743957008 T^6 - 1880399068065596812679449750312116489 T^7 + 1211937495049324668386707923551814144 T^8 - 759866924055411176816609501610145824 T^9 + 466557599052858501899389873590498576 T^10 - 280965824140635821336538113950238208 T^11 + 165486490562715543623266844910996960 T^12 - 95328733468347624721143436596991728 T^13 + 53730737569188242850960902675061540 T^14 - 29631735433275573295736684905520448 T^15 + 15982002519220233506297359288643328 T^16 - 8426732734596962888735943308790072 T^17 + 4341578043750972227945942898034432 T^18 - 2184193663643426076323203313845088 T^19 + 1072045107586559381111681621669072 T^20 - 512897616845631175409335289338708 T^21 + 239007878643078614755697662563584 T^22 - 108415793383957757795350567428064 T^23 + 47846270482094728117141329426032 T^24 - 20533661180243125068599265318144 T^25 + 8564906198781819799124804441280 T^26 - 3470264291680473250164651552944 T^27 + 1364870535759255877272510765950 T^28 - 520676891296255096870756895040 T^29 + 192488968788190123648373004064 T^30 - 68893036110679144584159460492 T^31 + 23845858487001866959614915840 T^32 - 7973063091544280406837942464 T^33 + 2572118763623299179804574640 T^34 - 799578831968317708137874814 T^35 + 239196982314145129630174464 T^36 - 68763448836715397230901728 T^37 + 18967378806716848507574128 T^38 - 5011787964028065103857408 T^39 + 1266306625250424841996640 T^40 - 305348843999288091901136 T^41 + 70117811645069434371412 T^42 - 15296768944400171831616 T^43 + 3162019501419003256064 T^44 - 617525327585232743224 T^45 + 113570706028361676288 T^46 - 19599347048769496032 T^47 + 3161153679144274672 T^48 - 474387152691155748 T^49 + 65902567592614400 T^50 - 8426269030832672 T^51 + 984947439372048 T^52 - 104425099694592 T^53 + 9947578647040 T^54 - 841756889488 T^55 + 62385936393 T^56 - 3978343968 T^57 + 213336304 T^58 - 9328642 T^59 + 318464 T^60 - 7936 T^61 + 128 T^62 - T^63 == 0, T, Reals, WorkingPrecision -> 200][[1]][[1]]][[1]]

A118746 Decimal expansion of onset of logistic map 7-bifurcation.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Apr 28 2006

Algebraic of order 114.

			3.7016407641603495818...

Eric Weisstein's World of Mathematics, Logistic Map

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

A118454 Algebraic degree of the onset of the logistic map n-bifurcation.

Original entry on oeis.org

1, 1, 2, 2, 22, 40, 114, 12, 480, 944, 2026, 3918, 8166, 16104, 32630, 240, 131038, 260928, 524250, 1046418, 2096706, 4190168, 8388562, 16768200, 33554240, 67092432, 134216136, 268402446, 536870854, 1073672968, 2147483586, 65280, 8589928346, 17179606976, 34359737478

Offset: 1

Eric W. Weisstein, Apr 28 2006

nonn

a(2^n) is A087046(n).

			The onsets begin at 1, 3, 1+2*sqrt(2), 1+sqrt(6), ...

Cheng Zhang, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Logistic Map

Cf. A000740, A006876, A086178, A086180, A087046, A091517, A098587, A118452, A118453, A118746, A181906, A181907, A181909, A181910, A181911.

degRp[n_] := Sum[MoebiusMu[n/d] 2^(d - 1), {d, Divisors[n]}]; degRo[n_] := degRp[n]*2 - Sum[EulerPhi[n/d] degRp[d], {d, Divisors[n]}]; Table[If[n <= 2, 1, 2 If[2^Round[Log2[n]] == n, degRp[n/2], degRo[n]]], {n, 1, 35}] (* Cheng Zhang, Apr 02 2012 *)

More terms from Cheng Zhang, Apr 02 2012

A087046 Algebraic order of r_n, the value of r in the logistic map that corresponding to the onset of the period 2^n-cycle.

Original entry on oeis.org

1, 2, 12, 240, 65280, 4294901760, 18446744069414584320, 340282366920938463444927863358058659840, 115792089237316195423570985008687907852929702298719625575994209400481361428480

Offset: 1

Eric W. Weisstein, Aug 04 2003

Ilias S. Kotsireas and Kostas Karamanos, Exact computation of the bifurcation point B4 of the logistic map and the Bailey-Broadhurst conjectures, Internat. J. Bifur. Chaos Appl. Sci. Engrg., Vol. 14, No. 7 (2004), pp. 2417-2423; alternative link.
Eric Weisstein's World of Mathematics, Logistic Map.

Cf. A086180, A086181, A087089, A118454, A000215, A346192.

Cf. A051179 (partial sums).

Table[If[n <= 1, 1, 2^(2^(n - 1)) - 2^(2^(n - 2))], {n, 1, 10}] (* Cheng Zhang, Apr 02 2012 *)

a(n) = 1<<(1<<(n-1)) - 1<<(1<<(n-2)); \\ Kevin Ryde, Jan 18 2024

a(n) = 2^(2^(n - 1)) - 2^(2^(n - 2)) with n>1, a(1)=1. - Cheng Zhang, Apr 02 2012

Sum_{n>=1} 1/a(n) = 1 + A346192. - Amiram Eldar, Jul 18 2021

More terms from Cheng Zhang, Apr 03 2012

A176213 Decimal expansion of 2+sqrt(6).

Original entry on oeis.org

4, 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, 8, 5, 9

Offset: 1

Klaus Brockhaus, Apr 12 2010

Continued fraction expansion of 2+sqrt(6) is A010694.

a(n) = A010464(n) = A086180(n) for n > 1, a(1) = 4.

			2+sqrt(6) = 4.44948974278317809819...

Cf. A010464 (decimal expansion of sqrt(6)), A086180 (decimal expansion of 1+sqrt(6)), A010694 (repeat 4, 2).

RealDigits[2+Sqrt[6],10,120][[1]] (* Harvey P. Dale, Aug 19 2018 *)

A379469 Decimal expansion of (1 + sqrt(6))/(3*sqrt(2)).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Dec 23 2024

This constant gives an upper bound to the Steiner ratio of a regular tetrahedron.

			0.81305252958514160597609690120357380207588039716628...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.6, p. 505.

Ding-Zhu Du and Warren D. Smith, Disproofs of Generalized Gilbert-Pollak Conjecture on the Steiner Ratio in Three or More Dimensions, Journal of Combinatorial Theory, Series A, Volume 74, Issue 1, 1996, Pages 115-130. See p. 128.
Warren D. Smith, How to find Steiner minimal trees in euclidean d-space, Algorithmica 7, 137-177 (1992).

Cf. A002193, A010464, A010474, A086180.

RealDigits[(1+Sqrt[6])/(3Sqrt[2]),10,100][[1]]

Equals A086180/A010474.

Minimal polynomial: 324*x^4 - 252*x^2 + 25. - Stefano Spezia, Aug 03 2025

cp's OEIS Frontend

A181918 The value of r at the bifurcation point of the first period-12 cycle of the logistic map f(x) = r*x*(1 - x).

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A181919 The value of r at the bifurcation point of the first period-13 cycle of the logistic map f(x) = r*x*(1 - x).

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A181912 The value of r at the bifurcation point of the first period-5 cycle of the logistic map f(x) = r*x*(1 - x).

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A181913 The value of r at the bifurcation point of the first period-7 cycle of the logistic map f(x) = r*x*(1 - x).

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A118746 Decimal expansion of onset of logistic map 7-bifurcation.

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A118454 Algebraic degree of the onset of the logistic map n-bifurcation.

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A087046 Algebraic order of r_n, the value of r in the logistic map that corresponding to the onset of the period 2^n-cycle.

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A176213 Decimal expansion of 2+sqrt(6).

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A379469 Decimal expansion of (1 + sqrt(6))/(3*sqrt(2)).

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A181918 The value of r at the bifurcation point of the first period-12 cycle of the logistic map f(x) = rx(1 - x).

A181919 The value of r at the bifurcation point of the first period-13 cycle of the logistic map f(x) = rx(1 - x).

A181912 The value of r at the bifurcation point of the first period-5 cycle of the logistic map f(x) = rx(1 - x).

A181913 The value of r at the bifurcation point of the first period-7 cycle of the logistic map f(x) = rx(1 - x).