On the Sum of Reciprocal Tribonacci Numbers

Abstract

The Tribonacci Zeta functions are defined by ${\zeta }_T(s)=\sum _{k=1}^{\mathrm{\infty }}{T}_k^{-s}$. We discuss the partial infinite sum $\sum _{n=1}^{\mathrm{\infty }}{T}_k^{-s}$ for some positive integer $n$. We also consider the continued fraction expansion including Tribonacci numbers.