Generating integers from a linear combination of integers

This is a famous problem, sometimes called the coin problem of Frobenius.

If $n=2$ the answer to your question is known to be $N = t_1 t_2 - (t_1 + t_2) + 1$. A proof of this can be found in the answer to this recent question.

For three or more integers, there is no known closed-form solution for $N$. There are some bounds on the values of $N$ in the $n = 3$ case, as well as some algorithms for determining $N$. For more information and references, see the Wikipedia and MathWorld pages on the Frobenius coin problem.

Basically, the problem is considered solved when $n = 2$, partially solved (because of the bounds and algorithms) when $n = 3$, and unsolved when $n \geq 4$.

Update: Guy's Unsolved Problems in Number Theory says, "The case $n = 3$ was first solved explicitly by Selmer and Beyer, using a continued fraction algorithm." So I guess the $n=3$ case has been solved. I suppose you have would have to dig up their paper (it's in the MathWorld references) to see their solution.