Posted 01/01/2016

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This problem is proposed by Ran Pan. If you want to submit your problem, please click here.

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This problem is about Frobenius coin problems.

The coin problem is a problem that asks for the largest monetary amount that cannot be obtained using only coins of specified denominations.

The solution to this problem for a given set of coin denominations is called the Frobenius number of the set. The Frobenius number of a given set $\{c_1,c_2,\cdots, c_n\}$ is denoted by $F(c_1,\cdots,c_n)$. In other words, $F(c_1,\cdots,c_n)$ is the largest number that cannot be partitioned into parts using $\{c_1,c_2,\cdots, c_n\}$. For example, $F(2,5)=3$ and $F(3,8)=13$. In general, $F(a,b)=ab-a-b$.

Unfortunately, there is no general formula for $F(a,b,c)$.

Suppose $a(n)=F(2+n,3+n,5+n)$. For example, $a(0)=F(2,3,5)=1$ and $a(1)=F(3,4,6)=5$.

Find $a(n)$.