Closed form of a recurrence formula

Question

I asked this question earlier by I was told that the statement was not clear so I tried to reword it.

Basically I am given a grid Ai(m x n), for any i natural, I am given the first terms on how the grid grows and I want to generalize it and find a formula for $i = n$ I was able to find a recursive definition but I am not sure how to convert it to a closed form.

First terms of the grid $$A0(1 X 1)$$ $$A1(1 X 3)$$ $$A2(3 X 3)$$ $$A3(3 X 7)$$ $$A4(7 X 7)$$ $$....$$

I got to the point where I deduced that:

If n even: The base case is for A(0) = 1, A(1) = 1 and $a(n+2) = 2a(n) +1$

For n odd: The base case is A(0) = 1, A(1) = 3 and $a(n+2) = 2a(n) +1$

so that I don't know how to proceed and get a get closed-form that would represent my problem.

Answer 1

The solution of the recurrence $a(n+2)=2 a(n)+1$ with $a(0)=b$ and $a(1)=c$ can be written as $$ a(n) = -1 + (1+(-1)^n) (b+1) 2^{n/2-1} + (1-(-1)^n) (c+1) 2^{n/2 - 3/2}$$