Generalized Tribonacci Function and Tribonacci Numbers

Abstract

In the language of mathematics, sequence is considered to be list of numbers arranged in a particular way. A lot of sequences have been minutely studied till date. One of the most conspicuous among them is Fibonacci sequence. It is the sequence, which can be found by adding two previous terms, where the initial conditions are 0 and 1. In a similar manner, Tribonacci sequence is also obtained by adding three previous consecutive terms. In this research paper, we introduce Tribonacci function 𝝓:ℝ→ℝ with period s (positive integer) such that 𝝓(𝒚+𝟑𝒔)=𝝓(𝒚+𝟐𝒔)+𝝓(𝒚+𝒔)+𝝓(𝒚),∀ 𝒚∈ℝ We construct some of the interesting properties, using induction technique, 𝝓 – odd function and 𝝓 - even function for Tribonacci function with period s. In the present research article we also show that 𝒍𝒊𝒎𝒚→∞𝝓(𝒚+𝒔)𝝓(𝒚) exists.