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

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 $a,bandc$ 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 $F_{i}x+F_{i+1}y+F_{i+2}z=n$ and $L_{i}x+L_{i+1}y+L_{i+2}z=n$ where i is fixed.