Systematic method, to find formula of algorithmic development I have an algorithm, to calculate the solution of a problem. I'd like to express the solution found by this algorithm, in term of the parameter of the problem. Furthermore, I'd like to find a method, that I could apply for this kind of problems.
Example:
I have a 2D matrix. The left and bottom border are filled with one's. Any other field is the sum of the left, left-bottom and bottom number. I'd like to find the number at the (n, n) indices.
1   9 41 129 321
1   7 25  63 129
1   5 13  25  41
1   3  5   7   9
1   1  1   1   1

By constructing the above matrix, I find that the solution of n=5 is 321. Constructing the algorithm that find the solution, is quite easy. But I'm at loss to find the algebraic formula. Despite this problem being very similar to Newton binomial, I don't see what kind of method I could use.
At the moment, I'd try to guess a function $f(n)$, and try to match $a\cdot f(n)+b$ using a numeric method. But guessing is not a systematic method...

I have the following... :


*

*Note the first row and column not units, is $2n+1$; so our next diagonal number is $$2n+1+2n+1+2(n-1)+1=6n+1$$

*Note our $n$ is even allowing us to replane $n$ with $2n$ getting $12n+1$ giving $$12(n-1)+1+24n+1+24n+1=72n-9$$

*Then it gets a bit weird. It doesn't hit 405 like iterating for odds would expect.  

