Geodesic
On the number of representations of an integer by a linear form
Journal of integer sequences, Tome 8 (2005) no. 5
Let $ a_1,\ldots,a_k$ be positive integers generating the unit ideal, and $ j$ be a residue class modulo $ L = \operatorname{lcm}(a_1,\ldots,a_k)$. It is known that the function $ r(N)$ that counts solutions to the equation $ x_1a_1 + \ldots + x_ka_k = N$ in non-negative integers $ x_i$ is a polynomial when restricted to non-negative integers $ N \equiv j \pmod L$. Here we give, in the case of $ k=3$, exact formulas for these polynomials up to the constant terms, and exact formulas including the constants for $ \mathfrak{q}= \gcd(a_1,a_2) \cdot \gcd(a_1,a_3) \cdot \gcd(a_2,a_3)$ of the $ L$ residue classes. The case $ \mathfrak{q}= L$ plays a special role, and it is studied in more detail.
Classification : 05A15, 52C07
Keywords: Frobenius problem, quasi-polynomial, representation numbers, pick's theorem 
@article{JIS_2005__8_5_a2,
     author = {Alon,  Gil and Clark,  Pete L.},
     title = {On the number of representations of an integer by a linear form},
     journal = {Journal of integer sequences},
     year = {2005},
     volume = {8},
     number = {5},
     zbl = {1108.11026},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2005__8_5_a2/}
}
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Alon,  Gil; Clark,  Pete L. On the number of representations of an integer by a linear form. Journal of integer sequences, Tome 8 (2005) no. 5. http://geodesic.mathdoc.fr/item/JIS_2005__8_5_a2/