New Tribonacci Recurrence Relations and Addition Formulas

Kunle Adegoke; Adenike Olatinwo; Winning Oyekanmi

paper

New Tribonacci Recurrence Relations and Addition Formulas

arXiv:1811.03663

Abstract

Only one three-term recurrence relation, namely, $W_{r} = 2 W_{r - 1} - W_{r - 4}$ , is known for the generalized Tribonacci numbers, $W_{r}$ , $r \in Z$ , defined by $W_{r} = W_{r - 1} + W_{r - 2} + W_{r - 3}$ and \mbox{ $W_{- r} = W_{- r + 3} - W_{- r + 2} - W_{- r + 1}$ }, where $W_{0}$ , $W_{1}$ and $W_{2}$ are given, arbitrary integers, not all zero. Also, only one four-term addition formula is known for these numbers, which is, $W_{r + s} = T_{s - 1} W_{r - 1} + (T_{s - 1} + T_{s - 2}) W_{r} + T_{s} W_{r + 1}$ , where $(T_{r})_{r \in Z}$ is the Tribonacci sequence, a special case of the generalized Tribonacci sequence, with $W_{0} = T_{0} = 0$ and $W_{1} = W_{2} = T_{1} = T_{2} = 1$ . In this paper we discover three new three-term recurrence relations and two identities from which a plethora of new addition formulas for the generalized Tribonacci numbers may be discovered. We obtain a simple relation connecting the Tribonacci numbers and the Tribonacci-Lucas numbers. Finally, we derive quadratic and cubic recurrence relations for the generalized Tribonacci numbers.

8 pages, no figures