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a(1)=1; a(n)=floor(4*a(n-1)*(n-a(n-1)) / n) for n > 1.
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%I #12 Dec 29 2023 10:36:13

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

%C 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 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 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 Harvey P. Dale, <a href="/A140357/b140357.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LogisticMap.html">Logistic Map.</a>

%t 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 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 Cf. A079271, A087089, A098587, A118454.

%K easy,nonn

%O 1,2

%A _Franklin T. Adams-Watters_, May 30 2008, May 31 2008