Let f_n be the standard Fibonacci sequence with f_0 = f_1 = 1. Define un as (1 + f_n), if n is even, and (n + f_n) if n is odd.
Find the generating function of u_n and then evaluate lim (n-> infinity) (u_(n+1))/ (u_n).
Here i defined v_n = 1 + f_n . Generating function of v_n is g(x)= 1/(1 - x - x^2) + 1/(1 - x). I also defined w_n = n + f_n and generating function of w_n is h(x) = 1/(1 - x - x^2) + x/(1 - x)^2.
Then generating function of u_n is f(x) = (1/2)*[ g(x) + g(-x) + h(x) - h(-x)].
Is this correct? If there is a mistake, then where is it?