2018/11/06 by Adegoke, Kunle, Olatinwo, Adenike, Oyekanmi, Winning
#11B37 #11B39 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1811.03663
Only one three-term recurrence relation, namely, Wr=2Wr-1-Wr-4, is known for the generalized Tribonacci numbers, Wr, r∈ℤ, defined by Wr=Wr-1+Wr-2+Wr-3 and \mboxW-r=W-r+3-W-r+2-W-r+1, where W0, W1 and W2 are given, arbitrary integers, not all zero. Also, only one four-term addition formula is known for these numbers, which is, Wr + s = Ts - 1 Wr - 1 + (Ts - 1 + Ts-2 )Wr + Ts Wr + 1, where (Tr)r∈ℤ is the Tribonacci sequence, a special case of the generalized Tribonacci sequence, with W0=T0=0 and W1=W2=T1=T2=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.