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A Fast Algorithm for Denumerants with Three Variables

Feihu Liu, Guoce Xin

Abstract

Let $a,b,c$ be distinct positive integers such that $a<b<c$ and $\gcd(a,b,c)=1$. For any non-negative integer $n$, the denumerant function $d(n;a,b,c)$ denotes the number of solutions of the equation $ax_1+bx_2+cx_3=n$ in non-negative integers $x_1,x_2,x_3$. We present an algorithm that computes $d(n;a,b,c)$ with a time complexity of $O(\log b)$.

A Fast Algorithm for Denumerants with Three Variables

Abstract

Let be distinct positive integers such that and . For any non-negative integer , the denumerant function denotes the number of solutions of the equation in non-negative integers . We present an algorithm that computes with a time complexity of .

Paper Structure

This paper contains 10 sections, 7 theorems, 43 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. A Brief Introduction to the Constant Term Method
  3. Several Reductions
  4. Reduction to the Case $\gcd(a,b)=1$
  5. Reduction to Two Contribution Term
  6. Reduction to the Case $b=1$
  7. A Euclid Style Recursive Operation
  8. Dispelling the Slack Variables
  9. The Algorithm
  10. Concluding Remark

Key Result

Proposition 2.1

Let $E(\lambda)$ be defined as in e-Elambda, where $z_i$ are complex parameters and the denominator factors are coprime to each other. Then $E(\lambda)$ has a partial fraction decomposition given by where $P(\lambda), p(\lambda)$, and the $A_i(\lambda)$'s are all polynomials, $\deg p(\lambda)<m$, and $\deg A_{i}(\lambda)<a_i$ for all $i$. Furthermore, the polynomial $A_i(\lambda)$ is uniquely cha

Theorems & Definitions (12)

  • Proposition 2.1: Xin15
  • Corollary 2.2
  • Theorem 2.3: XinZhang,The key transformation
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 4.1
  • proof
  • Example 4.2
  • ...and 2 more