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

1, 5, 14, 30, 66, 147, 316, 640, 1316, 2685, 5389, 10865, 21890, 43794, 87894, 176103, 352503, 705339, 1410939, 2822283, 5644683, 11290059, 22586380, 45177389, 90362673, 180726709, 361467845, 722962014, 1445926558, 2891903234

Offset: 1

Author

Ctibor O. Zizka, Oct 06 2009

Keywords

• nonn

Comments

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.

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

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

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.

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}.

Links

• Robert Israel, Table of n, a(n) for n = 1..2656
• J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.

Crossrefs

Cf. A006370, A048761

Programs

Maple

A[1]:= 1:
for n from 1 to 100 do
  A[n+1]:= A[n] + (floor(sqrt(A[n]))+1)^2
od:
seq(A[n],n=1..100); # Robert Israel, Oct 06 2014

PARI

lista(n) = {na = 0; for (i=1, n, na += ceil(sqrt(na+1))^2; print1(na, ", "););} \\ Michel Marcus, Jun 02 2013

Extensions

Typo in data corrected by D. S. McNeil, Aug 17 2010