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A264789
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Number of steps needed to reach 1 or to enter the cycle in the "sqrt(3)*x+1" problem.
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3
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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
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OFFSET
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1,4
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COMMENTS
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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, ...
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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Table[Length@ NestWhileList[If[EvenQ@ #, #/2, Floor[# Sqrt@ 3 + 1]] &, n, UnsameQ, All] - 2, {n, 0, 72}] (* Michael De Vlieger, Nov 25 2015 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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