cp's OEIS Frontend

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

%I A140357 #13 Feb 16 2025 08:33:08
%S A140357 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,
%T A140357 21,25,19,30,10,28,22,34,11,31,25,37,14,37,20,43,7,23,46,7,24,49,7,24,
%U A140357 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
%N A140357 a(1)=1; a(n)=floor(4*a(n-1)*(n-a(n-1)) / n) for n > 1.
%C A140357 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.
%C A140357 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.
%C A140357 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.
%H A140357 Harvey P. Dale, <a href="/A140357/b140357.txt">Table of n, a(n) for n = 1..1000</a>
%H A140357 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogisticMap.html">Logistic Map.</a>
%t A140357 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 *)
%t A140357 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 *)
%Y A140357 Cf. A079271, A087089, A098587, A118454.
%K A140357 easy,nonn
%O A140357 1,2
%A A140357 _Franklin T. Adams-Watters_, May 30 2008, May 31 2008