A086178 -id:A086178 - cp's OEIS frontend

A181915 The value of r at the bifurcation point of the first period-9 cycle of the logistic map f(x) = rx(1 - x).

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

Offset: 1

Cheng Zhang, Apr 01 2012

Root of a degree 252*2 = 504 polynomial.

			3.6872742105...

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

Cf. A086178, A086179, A086180, A086181, A091517, A118452, A118453, A118746, A181906, A181907, A181909, A181910, A181911, A181912, A181913, A181916, A181917, A181918, A181919.

A181916 The value of r at the bifurcation point of the first period-10 cycle of the logistic map f(x) = rx(1 - x).

Original entry on oeis.org

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

Offset: 1

Cheng Zhang, Apr 01 2012

Root of a degree 495*2 = 990 polynomial.

			3.605916932....

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

Cf. A086178, A086179, A086180, A086181, A091517, A118452, A118453, A118746, A181906, A181907, A181909, A181910, A181911, A181912, A181913, A181915, A181917, A181918, A181919.

A181917 The value of r at the bifurcation point 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, 2, 6, 6, 4, 5, 6, 5, 1, 7, 4, 3, 5, 6, 4, 8, 0, 5, 1, 4, 6, 6, 5, 6, 0, 4, 4, 1, 8, 2, 7, 5, 0, 4, 6, 2, 3, 4, 3, 9, 9, 4, 9, 2, 1, 3, 7, 4, 4, 6, 6, 1, 8, 3, 8, 0, 4, 4, 6, 8, 2, 8, 4, 3, 3, 8, 6, 3, 3, 7, 5, 9, 4, 1, 6, 0, 6, 7, 3, 5, 4, 7, 0, 0, 0, 4, 1, 0, 0, 9, 3, 4, 4, 2, 2, 4

Offset: 1

Cheng Zhang, Apr 01 2012

Root of a degree 1023*2 = 2046 polynomial.

			3.68172664565...

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

Cf. A086178, A086179, A086180, A086181, A091517, A118452, A118453, A118746, A181906, A181907, A181909, A181910, A181911, A181912, A181913, A181915, A181916, A181918, A181919.

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

Original entry on oeis.org

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

A163960 Decimal expansion of 2*(sqrt(2) - 1).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Oct 02 2010

Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(1,1,sqrt(2)). (See A195284.) - Clark Kimberling, Sep 14 2011

			0.82842712474619009760337744841939615713934375075389614635335...

J. M. Steele, Probability Theory and Combinatorial Optimization, SIAM, 1997, p. 3.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for algebraic numbers, degree 2.

Cf. A084158, A156035, A195284.

Essentially the same digit sequence as A010466, A086178, A090488 and A157258.

RealDigits[2(Sqrt[2]-1),10,120][[1]] (* Harvey P. Dale, May 27 2016 *)

2*(sqrt(2)-1) \\ G. C. Greubel, Aug 13 2017

Equals Sum_{k>=0} (-1)^k * binomial(2*k,k)/((k+1) * 4^k). - Amiram Eldar, May 06 2022

Equals Sum_{k>=1} (-1)^(k+1)/A084158(k). - Amiram Eldar, Dec 02 2024

cp's OEIS Frontend

A181915 The value of r at the bifurcation point of the first period-9 cycle of the logistic map f(x) = r*x*(1 - x).

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A181916 The value of r at the bifurcation point of the first period-10 cycle of the logistic map f(x) = r*x*(1 - x).

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A181917 The value of r at the bifurcation point of the first period-11 cycle of the logistic map f(x) = r*x*(1 - x).

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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) = 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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A163960 Decimal expansion of 2*(sqrt(2) - 1).

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A181915 The value of r at the bifurcation point of the first period-9 cycle of the logistic map f(x) = rx(1 - x).

A181916 The value of r at the bifurcation point of the first period-10 cycle of the logistic map f(x) = rx(1 - x).

A181917 The value of r at the bifurcation point of the first period-11 cycle of the logistic map f(x) = rx(1 - x).

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).