The Number of Solutions to $ax+by+cz=n$ for Fibonacci and Lucas triplets

TL;DR

This paper derives exact formulas for counting solutions to the linear equation with Fibonacci and Lucas triplet coefficients, extending previous work that used summations of floor functions.

Contribution

It provides the first explicit formulas for solutions when coefficients are consecutive Fibonacci or Lucas numbers, improving on prior summation-based formulas.

Findings

Exact formulas for Fibonacci triplet solutions derived

Exact formulas for Lucas triplet solutions derived

Extends previous summation-based methods to explicit formulas

Abstract

In this work we develop exact formulas to the number of solutions of $ax+by+cz=n$ in some special cases. In 2020, Binner gave a formula for the number of non negative integer solutions, $N(a,b,c;n)$ in non-negative integer pairs $(x,y,z),$ of the equation $ax+by+cz=n$ assuming that $a,b,c$ and $n$ are natural numbers. However, his formula was in summations of floor functions. Moreover, he gave a reciprocity relation to solve these sums by generalising the Gauss reciprocity relation. Until now no exact formula has been found to solve these sums. We notice that these sums can be completely solved in some special cases, which lead us to find the number of solutions of the above equation in case of Fibonacci and Lucas triplets. In other words; If $a,b\ and\ c$ are chosen to be three consecutive Fibonacci or Lucas numbers then we determine the exact formula to the number of non-negative integer solutions $(x,y,z)$ of the equation $F_ix+F_{i+1}y+F_{i+2}z=n$ and $L_ix+L_{i+1}y+L_{i+2}z=n$ where i is fixed.