1,9

For m = 1, the expression for x becomes x=(2^c[1] - 1) / 3.

Also the number of odd x with stopping time n for the Collatz or 3x+1 problem where x->x/2 if x is even, x->(3x+1)/2 if x is odd (see A060322), except that 1 is counted as having stopping time 2 instead of 0.

Equivalently, a(n) is the number of x == 2 (mod 3) with stopping time n-1.

The number of possible c[1],...,c[m] is 2^(n-1)-2^(n-3); most do not yield integer x.

n-c[m], n-c[m-1], ..., n-c[2] are the stopping times of the odd integers in the Collatz trajectory of x.

a(n) = a(n-2) + a(n-2):(x is 1 mod 6) + a(n-1):(x is 5 mod 6)

It is conjectured that a(n)/a(n-1) -> 4/3 as n-> infinity.

Perry Dobbie, Collatz representations.

Index entries for sequences related to 3x+1 (or Collatz) problem

For n=3, the only valid c are:

c=(3,2,1) (2^3 - 2^2 - 3^1*2^1 - 3^2) / 3^3 = -11/27,

c=(3,2) (2^3 - 2^2 - 3^1) / 3^2 = 1/9,

c=(3) (2^3 - 2^0 ) / 3 = 7/3,

and none are integers so a(3) = 0.

a(9)=2

c=(9,5) (2^9 - 2^5 - 3) / 3 = 53

c=(9,5,2) (2^9 - 2^5 - 3*2^2 - 9) / 27 = 17

and no other valid c give integer x.

Sequence in context: A111150 A166983 A078611 this_sequence A114218 A111973 A133691

Adjacent sequences: A131447 A131448 A131449 this_sequence A131451 A131452 A131453

nonn

Perry Dobbie (pdobbie(AT)rogers.com), Jul 11 2007, Jul 12 2007, Jul 13 2007, Jul 17 2007, Jul 22 2007, Oct 15 2008

Edited by David Applegate, Oct 16 2008