I'm trying to prove for every positive even number, $$x = 2y$$ that
$$\sum_{i=0}^y {x \choose 2i} 2^{2i} = \sum_{i=0}^{y-1} {x \choose 2i + 1}2^{2i+1} + 1$$
Also to find and prove a similar equation for odd positive numbers $$x = 2y+1$$
I'm trying to prove for every positive even number, $$x = 2y$$ that
$$\sum_{i=0}^y {x \choose 2i} 2^{2i} = \sum_{i=0}^{y-1} {x \choose 2i + 1}2^{2i+1} + 1$$
Also to find and prove a similar equation for odd positive numbers $$x = 2y+1$$