Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.2

On the Number of Representations of an Integer by a Linear Form

Gil Alon and Pete L. Clark
1126 Burnside Hall
Department of Mathematics and Statistics
McGill University
805 Sherbrooke West
Montreal, QC H3A 2K6


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.

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Received June 23 2005; revised version received October 19 2005. Published in Journal of Integer Sequences October 20 2005.

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