collatz 3x+1

[enzocreti]

I found a forumula about the 3x+1 problem


let be T(k,n) the trajectory starting from integer n.


k is the number of iterations


T(0,n)=n i think


T(k,n)=n*lambda(k,n)+rho(k,n) how to proof that?


where lamba(k,n)=(3^(x(0,n)+...+x(k-1,n))/2^k


x(0,n)...is the parity vector of 1's and 0's, 1 if x(i,n) is odd and 0 if it is even


rho is a sum from i=0 to k-1 of


x(i,n)*((3^(x(i+1),n)+...+x(k-1,n))/2^(k-i)



?