New Tribonacci recurrence relations and addition formulas

Kunle Adegoke, Adenike Olatinwo and Winning Oyekanmi
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367-8275
Volume 26, 2020, Number 4, Pages 164–172
DOI: 10.7546/nntdm.2020.26.4.164-172
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Authors and affiliations

Kunle Adegoke
Department of Physics and Engineering Physics, Obafemi Awolowo University
220005 Ile-Ife, Nigeria

Adenike Olatinwo
Department of Physics and Engineering Physics, Obafemi Awolowo University
220005 Ile-Ife, Nigeria

Winning Oyekanmi
Department of Physics and Engineering Physics, Obafemi Awolowo University
220005 Ile-Ife, Nigeria

Abstract

Only one three-term recurrence relation, namely, W_{r}=2W_{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 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.

Keywords

  • Tribonacci number
  • Tribonacci–Lucas number
  • Recurrence relation

2010 Mathematics Subject Classification

  • 11B39
  • 11B37

References

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Cite this paper

Adegoke, K., Olatinwo, A., & Oyekanmi, W. (2020). New Tribonacci recurrence relations and addition formulas. Notes on Number Theory and Discrete Mathematics, 26 (4), 164-172, DOI: 10.7546/nntdm.2020.26.4.164-172.

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