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A264789
Number of steps needed to reach 1 or to enter the cycle in the "sqrt(3)*x+1" problem.
3
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
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Nov 25 2015
STATUS
approved