I know it's a code review, but since you're chasing efficiency, here's how to calculate in O(1).  Try it first in Excel...

INT() is traditional TRUNC()/FLOOR() for positive integers  
Integer MOD(a,b) gives remainder after dividing a by b    

N is season number, h the height (at the end of that season)

    h = (2^(INT(N/2)+1)-1)*(1+INT(MOD(N,2)))

Output

    N   h
    0   1
    1   2
    2   3
    3   6
    4   7
    5  14
    6  15
    7  30