Let me define a recurence like this

S(1) = 1

S(2) = 1

S(n) = S(n-1) + S(n-2) + a; where a is some number

Q1: What is the relation beween S(n) and F(n) [nth Fibonacci number]

I found out this is S(n) = F(n) + K(n)*a

Where K(n) is another series of the type S(n) where a = 1

Q2: Is there a way to find the generic term for S(n) ?

Any pointers will be of great help. Thanks