PARTIAL SUMS OF THE 3X+1 ACCUMULATION TERM

The "3x+1" accumulation term, E(v_k), is what becomes of the "+1" part after k iterations of the "shortcut" Collatz function T(n) which takes odd integers n to (3n+1)/2 and even integers n to n/2, so T^k(n)=(3^i*n+E(v_k))/2 where i is the number of odd terms in the trajectory of n. A known and easy to prove result is that the total sum of the accumulation terms of all the 2^k possible parity vectors v_k of length k is k4^(k-1). This paper will show that the partial sum of the accumulation terms of all vectors v_k of length k for a fixed value of i is the sum from p=0 to i-1 of the binomial(k,p)*(2^k - 3^p). A very insightful proof of this will be presented, as well as a more classical one.