I am stuck with one formula at page 35 of The Ultimate Challange the 3x+1 problem.
$T^{(i)}(n)\equiv x_i(n) \pmod 2$
$x_i$ form the so called parity vector of the Collatz sequence.
Example of parity vector taken from E. Rosendaal page: For n= 17 we find v0 = 1, v1 = 0, v2 = 1, v3 = 0, v4 = 0, v5 = 1, etc.
Now
$T^{(i)}(n)=\lambda_k(n)\cdot n+\rho_k(n)$
where
$\lambda_k(n)=\frac {3^{x_0(n)}+...+3^{{k-1}(n)}}{2^k}$
and
$\rho_k(n)=\sum_{i=0}^{k-1} x_i(n) \frac{3^{x_{i+1}(n)+...+x_{k-1}(n)}}{2^{k-i}}$
T is the iteration function, n is the starting value of the sequence, i is the i-th iteration of the T function.
How to prove this formula?
I am completely stuck.