I was not able to find a closed form solution of f(a,b) that depends solely on the two values a and b.  I did find one that comes close and is correct for every value except the first line:  

if  the first line f(2,3) were 9 instead of 10, then that solution would be valid.  

And for that formula, the last line f(7,8) would be 99.  I suspect this is the same one found by tomarken.

The formula I did find, which depends on the Table is:

a^2 + ab - the value of "a" one line above.

This also gives the value of f(7,8) in the last line of 99.

In the case of the first line, there is no line above, so the third term is zero.