A Unified Explicit Binet Formula for 3rd-Order Linear Recurrence Relations

In this paper, third order generalized linear recurrence relation Vn (aj , pj) = p1Vn−1 + p2Vn−2 + p3Vn−3, p3 ≠ 0, is studied to generate a generalized Tribonacci sequence, where pj , Vj = aj are arbitrary integers. Generalized generating function for the 3rd order general tribonacci sequence is derived, and then new unified explicit generalized Binet formulas is obtained. This formula is then compared with existing ones. Furthermore, by imposing specific constraints on the initial terms and coefficients of the recurrence relation, the formula is specialized to the Fibonacci sequence and other Fibonacci-like sequences by appropriately selecting pj , Vj = aj.