A264789 Number of steps needed to reach 1 or to enter the cycle in the "sqrt(3)*x+1" problem.

0, 1, 1, 2, 6, 1, 12, 3, 5, 7, 9, 2, 11, 13, 13, 4, 15, 6, 17, 8, 9, 10, 10, 3, 12, 12, 12, 14, 17, 14, 14, 5, 17, 16, 16, 7, 8, 17, 18, 9, 9, 10, 11, 11, 20, 11, 11, 4, 8, 13, 17, 13, 13, 13, 6, 15, 15, 17, 49, 15, 15, 15, 8, 6, 8, 18, 17, 17, 17, 17, 44, 8

Offset

1,4

Comments

The sqrt(3)*x+1 problem is as follows: start with a number x. If x is even, divide it by 2, otherwise multiply it by sqrt(3) and add 1, and then take the integer part.

There are three possible behaviors for such trajectories when n>0:

(i) The trajectory reaches 1 (and enters the "trivial" cycle 2-1-2-1-2...).

(ii) Cyclic trajectory. The trajectory becomes periodic and the period does not contain a 1.

(iii) The trajectory is divergent trajectory (I conjecture that this cannot occur).

For many numbers, the element of the trivial cycle is 1, except for the numbers: 3, 6, 12, 19, 21, 24, 29, 33, 37, 38, 42, 43, 48, 49, 51, 55, 57, 58, ... where the elements of the nontrivial cycle are respectively 6, 3, 3, 38, 74, 3, 58, 19, 74, 76, 74, 37, 3, 98, 29, 6, 37, 33, ...

Links

Michel Lagneau, Table of n, a(n) for n = 1..10000

Example

a(3) = 1 because 3 -> 6 -> 3 -> 6 ...
a(7) = 12 because 7 -> 13 -> 23 -> 40 -> 20 -> 10 -> 5 -> 9 -> 16 -> 8 -> 4 -> 2 -> 1 where:
  13 = floor(7*sqrt(3)+1);
  23 = floor(13*sqrt(3)+1);
  40 = floor(23*sqrt(3)+1);
  20 = 40/2;
  10 = 20/2;
   5 = 10/2;
   9 = floor(5*sqrt(3)+1);
  16 = floor(9*sqrt(3)+1);
   8 = 16/2; 4 = 8/2; 2 = 4/2 and 1 = 2/2 is the end of the cycle.

Maple

A264789 := proc(n)
    local cyc, x;
    x := n;
    cyc := {x} ;
    for s from 0 do
        if 1 in cyc then
            return s;
        end if;
        if type(x, 'even') then
            x := x/2 ;
        else
            x := floor(sqrt(3)*x+1) ;
        end if;
        if x in cyc and s > 0 then
            return s;
        end if;
        cyc := cyc union {x} ;
    end do:
end proc: # R. J. Mathar, Nov 27 2015

Mathematica

Table[Length@ NestWhileList[If[EvenQ@ #, #/2, Floor[# Sqrt@ 3 + 1]] &, n, UnsameQ, All] - 2, {n, 0, 72}] (* Michael De Vlieger, Nov 25 2015 *)

Crossrefs

Cf. A006577.

Sequence in context: A100831 A136763 A109530 * A111519 A184941 A185340

Adjacent sequences: A264786 A264787 A264788 * A264790 A264791 A264792

Keyword

nonn

Author

Michel Lagneau, Nov 25 2015

Status

approved