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A166068 a(n) = a(n-1)+ [least square > a(n-1)]. 1

%I

%S 1,5,14,30,66,147,316,640,1316,2685,5389,10865,21890,43794,87894,

%T 176103,352503,705339,1410939,2822283,5644683,11290059,22586380,

%U 45177389,90362673,180726709,361467845,722962014,1445926558,2891903234

%N a(n) = a(n-1)+ [least square > a(n-1)].

%C This sequence is the base sequence of the map: a(n) = a(n-1)+ [least square > a(n-1)] if a(n) is not divisible by Y, else a(n)=a(n-1)/Y, where Y is a positive integer.

%C Experimental results shows this map converges to a periodic orbit for all Y.

%C What is the number and length of periodic orbits for different Y?

%C What is the trajectory of some input under the map? If Y=2, the map converges to two periodic orbits, {1-5-14-7-16-8-4-2} and {11-27-63-127-271-560-280-140-70-35-71-152-76-38-19-44-22} whose length is L1=8, L2=17.

%C Two examples of trajectories for initial value 9 resp. 13 under the map for Y=2 are 9-25-61-125-269-558-279-568-284-142-{76-38-19-44-22-11-27-63-127-271-560-280-140-70-35-71-152} and 13-29-65-146-73-154-77-158-79-160-80-40-20-10-{5-14-7-16-8-4-2-1}.

%H Robert Israel, <a href="/A166068/b166068.txt">Table of n, a(n) for n = 1..2656</a>

%H J. C. Lagarias, <a href="http://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/paper.html">The 3x+1 problem and its generalizations</a>, Amer. Math. Monthly, 92 (1985), 3-23.

%p A[1]:= 1:

%p for n from 1 to 100 do

%p A[n+1]:= A[n] + (floor(sqrt(A[n]))+1)^2

%p od:

%p seq(A[n],n=1..100); # _Robert Israel_, Oct 06 2014

%o (PARI) lista(n) = {na = 0; for (i=1, n, na += ceil(sqrt(na+1))^2; print1(na, ", "););} \\ _Michel Marcus_, Jun 02 2013

%Y Cf. A006370, A048761

%K nonn

%O 1,2

%A _Ctibor O. Zizka_, Oct 06 2009

%E Typo in data corrected by _D. S. McNeil_, Aug 17 2010

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Last modified April 16 12:45 EDT 2021. Contains 343037 sequences. (Running on oeis4.)