Frobenius coin problem
Given a list of positive integers $a_1,\ldots,a_n$, the Frobenius coin problem asks for the largest integer $N$ such that the equation $$ x_1 a_1 + \cdots + x_n a_n = N $$ has no solutions where $x_i$ are nonnegative integers. This integer is denoted as $g(a_1,\ldots,a_n)$.
A related question asks whether a given $N$ has such a solution. A necessary (albeit not sufficient) condition is that $N$ is divisible by $G = \mathrm{gcd}(a_1,\ldots,a_n)$. However, it can be shown that, if $A = \max_i a_i$, any $N>A^2$ has such a solution iff. $N$ is divisible by $G$.need Thus the Frobenius coin problem is only well defined when $G=1$.
These problems are NP-Hard, although they can be solved in pseudo-polynomial time $O(na_1\log(na_1))$ 12
Identities for small $n$
$n=2$
There is a closed-form solution for $n=2$: $$ g(a_1,a_2) = a_1a_2 - a_1 - a_2 $$ Furthermore, the number of integers that don't have a solution is given by the formula: $$ N(a_1,a_2) = (a_1-1)(a_2-1)/2 $$
$n=3$
There is no known closed-form solution for $n=3$, although the following equality is known to hold:need $$ g(d\cdot a_1, d\cdot a_2, a_3) = d\cdot g(a_1,a_2,a_3) + a_3(d-1) $$