ϕ(x)-Tribonnaci polynomial, numbers, and its sum

This study presents a general third-order nabla difference operator that allows us to get ϕ(x)-Tribonacci sequences, Tri-bonacci numbers, and their sum using the coefficients of different trigonometric functions and their inverse. In this section, we examined the numerical solutions and C^∗-solutions of the ϕ(x)-Tribonacci sequences for different functions. In addition, some interesting conclusions and theorems are obtained for the sum of the terms of the Tribonacci sequence. Also, we offer appropriate examples to show how to use MATLAB to demonstrate our results.