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

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

Author

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

Keywords

• easy
• nonn

Comments

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.

Links

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

Crossrefs

Cf. A079271, A087089, A098587, A118454.

Programs

Mathematica

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 *)