A087089 -id:A087089 - cp's OEIS frontend

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.

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

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

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cp's OEIS Frontend

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.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

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Mathematica

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