A118454 -id:A118454 - cp's OEIS frontend

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

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]]

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]]

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.

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

A006876 Mu-molecules in Mandelbrot set whose seeds have period n.

Original entry on oeis.org

1, 0, 1, 3, 11, 20, 57, 108, 240, 472, 1013, 1959, 4083, 8052, 16315, 32496, 65519, 130464, 262125, 523209, 1048353, 2095084, 4194281, 8384100, 16777120, 33546216, 67108068, 134201223, 268435427, 536836484, 1073741793, 2147417952, 4294964173, 8589803488, 17179868739

Offset: 1

Robert Munafo

nonn

B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, NY, 1982, p. 183.
R. Penrose, The Emperor's New Mind, Penguin Books, NY, 1991, p. 138.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Cheng Zhang, Table of n, a(n) for n = 1..1000
R. P. Munafo, Enumeration of Features

Cf. A006874, A006875, A000740, A118454.

degRp[n_] := Sum[MoebiusMu[n/d] 2^(d - 1), {d, Divisors[n]}]; Table[degRp[n]*2 - Sum[EulerPhi[n/d] degRp[d], {d, Divisors[n]}], {n, 1, 100}] (* Cheng Zhang, Apr 02 2012 *)

A000740(n)=sumdiv(n,d,moebius(n/d)<<(d-1))
a(n)=2*A000740(n)-sumdiv(n, d, eulerphi(n/d)*A000740(d)) \\ Charles R Greathouse IV, Feb 18 2013

a(n) = 2*l(n) - sum_{d|n} phi(n/d)*l(d), where l(n) = sum_{d|n} mu(n/d) 2^(d-1) (A000740), and phi(n) and mu(n) are the Euler totient function (A000010) and Moebius function (A008683), respectively. - Cheng Zhang, Apr 02 2012

Web link changed to more relevant page by Robert Munafo, Nov 16 2010

More terms from Cheng Zhang, Apr 02 2012

A006875 Non-seed mu-atoms of period n in Mandelbrot set.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 12, 12, 23, 10, 51, 12, 75, 50, 144, 16, 324, 18, 561, 156, 1043, 22, 2340, 80, 4119, 540, 8307, 28, 17521, 30, 32928, 2096, 65567, 366, 135432, 36, 262179, 8250, 525348, 40, 1065093, 42, 2098263, 33876, 4194347, 46, 8456160, 420, 16779280

Offset: 1

Robert Munafo

nonn

Definitions and Maxima source code on second Munafo web page. - Robert Munafo, Dec 12 2009

			From _Robert Munafo_, Dec 12 2009: (Start)
For n=1 the only mu-atom is the large cardioid, which is a seed.
For n=2 there is one, the large circular mu-atom centered at -1+0i, so a(2)=1.
For n=3 there is a seed (cardioid) at -1.75+0i, which doesn't count, and two non-seeds ("circles") at approx. -0.1225+-0.7448i, so a(3)=2. (End)

B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, NY, 1982, p. 183.
R. Penrose, The Emperor's New Mind, Penguin Books, NY, 1991, p. 138.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Indranil Ghosh, Table of n, a(n) for n = 1..1000
R. P. Munafo, Mu-Ency - The Encyclopedia of the Mandelbrot Set
R. Munafo, Enumeration of Features [From _Robert Munafo_, Dec 12 2009]

Cf. A000740, A006874, A006876, A118454.

Table[Sum[EulerPhi[n/d] Sum[MoebiusMu[d/c] 2^(c - 1), {c, Divisors[d]}], {d, Drop[Divisors[n], -1]}], {n, 1, 100}] (* Cheng Zhang, Apr 03 2012 *)

from sympy import divisors, totient, mobius
l=[0, 0]
for n in range(2, 101):
    l.append(sum(totient(n//d)*sum(mobius(d//c)*2**(c - 1) for c in divisors(d)) for d in divisors(n)[:-1]))
print(l[1:]) # Indranil Ghosh, Jul 12 2017

a(n) = Sum_{d|n, d < n} (phi(n/d) * sum_{c|d} (mu(d/c) 2^(c-1))), where phi(n) and mu(n) are the Euler totient function (A000010) and Moebius function (A008683), respectively. - Cheng Zhang, Apr 03 2012

a(n) = A000740(n) - A006876(n).

A140357 a(1)=1; a(n)=floor(4a(n-1)(n-a(n-1)) / n) for n > 1.

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 3, 7, 6, 9, 6, 12, 3, 9, 14, 7, 16, 7, 17, 10, 20, 7, 19, 15, 24, 7, 20, 22, 21, 25, 19, 30, 10, 28, 22, 34, 11, 31, 25, 37, 14, 37, 20, 43, 7, 23, 46, 7, 24, 49, 7, 24, 52, 7, 24, 54, 11, 35, 56, 14, 43, 52, 36, 63, 7, 25, 62, 21, 58, 39, 70, 7, 25, 66, 31, 73, 15, 48

Offset: 1

Franklin T. Adams-Watters, May 30 2008, May 31 2008

a(n)/n approximates the behavior of the logistic map x(n+1) = r*x(n)*(1-x(n)) at the critical value r = 4 where its iterated behavior becomes chaotic.
Conjecture: starting with any given n and any 1 <= a(n) <= n and applying the rule for the sequence produces a sequence which eventually joins this one. For example, starting with a(9)=5, the sequence continues 10,3,9,11,9, at which point it has joined.
There is a number x(1) such that iterating the logistic map x(n+1) = 4*x(n)*(1-x(n)) approaches a(n)/n; in particular x(n) > 1/2 iff a(n)/n > 1/2 and lim_{n->infinity} x(n)-a(n)/n = 0. x(1) is approximately 0.74300456748016924159182578873962328734252790178266693834898117732270042549583799064232908893034253248. It appears that |x(n)-a(n)/n| < 1/sqrt(n) for all n.

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

Cf. A079271, A087089, A098587, A118454.

a[1]=1;a[n_]:=a[n]=Floor[(4a[n-1](n-a[n-1]))/n];Table[a[n],{n,100}]  (* Harvey P. Dale, Mar 28 2011 *)
nxt[{n_,a_}]:={n+1,Floor[4a (n+1-a)/(n+1)]}; NestList[nxt,{1,1},80][[All,2]] (* Harvey P. Dale, Dec 22 2019 *)

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

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

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A118746 Decimal expansion of onset of logistic map 7-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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A006876 Mu-molecules in Mandelbrot set whose seeds have period n.

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A006875 Non-seed mu-atoms of period n in Mandelbrot set.

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A140357 a(1)=1; a(n)=floor(4*a(n-1)*(n-a(n-1)) / n) for n > 1.

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A140357 a(1)=1; a(n)=floor(4a(n-1)(n-a(n-1)) / n) for n > 1.