Abstract
In the present paper, for a positive integer $r$, we study bi-periodic $r$-Fibonacci sequence and its family of companion sequences, bi-periodic $r$-Lucas sequence of type $s$ with $1 \leq s \leq r$, which extend the classical Fibonacci and Lucas sequences. Afterwards, we establish the link between the bi-periodic $r$-Fibonacci sequence and its companion sequences. Furthermore, we give their properties as linear recurrence relations, generating functions, explicit formulas and Binet forms.
Keywords
bi-periodic r-Fibonacci sequence bi-periodic r-Lucas sequence companion sequence recurrence relation generating function explicit formula Binet form
References
- [1] S. Abbad, H. Belbachir and B. Benzaghou, Companion sequences associated to the r-Fibonacci sequence: algebraic and combinatorial properties, Turk. J. Math. 43 (3), 1095-1114, 2019.
- [2] H. Belbachir, A combinatorial contribution to the multinomial Chu-Vandermonde convolution, Les Annales RECITS 1, 27-32, 2014.
- [3] H. Belbachir and F. Bencherif, Linear recurrent sequences and powers of a square matrix, Integers 6, A12, 2006.
- [4] G. Bilgici, Two generalizations of Lucas sequence, Appl. Math. Comput. 245, 526- 538, 2014.
- [5] L. Cerlienco, M. Mignotte and F. Piras, Suites récurrentes linéaires, propriétés algébriques et arithmétiques, Enseignement Mathématiques 33, 67-108, 1987.
- [6] M. Edson and O. Yayenie, A new generalization of Fibonacci sequences and extended Binet’s Formula, Integers, 9, 639-654, 2009.
- [7] D. Kalman, Generalized Fibonacci Numbers by matrix methods, Fibonacci Quart. 20 (1), 73-76, 1982.
- [8] J.A. Raab A generalization of the connection between the Fibonacci sequence and Pascal’s triangle, Fibonacci Quart. 1, 21-31, 1963.
- [9] M. Sahin, The Gelin-Cesàro identity in some conditional sequences, Hacet. J. Math. Stat. 40 (6), 855-861, 2011.
- [10] E. Tan and A.B. Ekin, Bi-periodic Incomplete Lucas Sequences, Ars Combin. 123, 371-380, 2015.
- [11] O. Yayenie, A note on generalized Fibonacci sequence, Appl. Math. Comput. 217, 5603-5611, 2011.
- [12] Y. Yazlik, C. Köme and V. Madhusudanan, A new generalization of Fibonacci and Lucas p-numbers, J. Comput. Anal. Appl. 25 (4), 657-669, 2018.
Abstract
References
- [1] S. Abbad, H. Belbachir and B. Benzaghou, Companion sequences associated to the r-Fibonacci sequence: algebraic and combinatorial properties, Turk. J. Math. 43 (3), 1095-1114, 2019.
- [2] H. Belbachir, A combinatorial contribution to the multinomial Chu-Vandermonde convolution, Les Annales RECITS 1, 27-32, 2014.
- [3] H. Belbachir and F. Bencherif, Linear recurrent sequences and powers of a square matrix, Integers 6, A12, 2006.
- [4] G. Bilgici, Two generalizations of Lucas sequence, Appl. Math. Comput. 245, 526- 538, 2014.
- [5] L. Cerlienco, M. Mignotte and F. Piras, Suites récurrentes linéaires, propriétés algébriques et arithmétiques, Enseignement Mathématiques 33, 67-108, 1987.
- [6] M. Edson and O. Yayenie, A new generalization of Fibonacci sequences and extended Binet’s Formula, Integers, 9, 639-654, 2009.
- [7] D. Kalman, Generalized Fibonacci Numbers by matrix methods, Fibonacci Quart. 20 (1), 73-76, 1982.
- [8] J.A. Raab A generalization of the connection between the Fibonacci sequence and Pascal’s triangle, Fibonacci Quart. 1, 21-31, 1963.
- [9] M. Sahin, The Gelin-Cesàro identity in some conditional sequences, Hacet. J. Math. Stat. 40 (6), 855-861, 2011.
- [10] E. Tan and A.B. Ekin, Bi-periodic Incomplete Lucas Sequences, Ars Combin. 123, 371-380, 2015.
- [11] O. Yayenie, A note on generalized Fibonacci sequence, Appl. Math. Comput. 217, 5603-5611, 2011.
- [12] Y. Yazlik, C. Köme and V. Madhusudanan, A new generalization of Fibonacci and Lucas p-numbers, J. Comput. Anal. Appl. 25 (4), 657-669, 2018.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Research Article |
| Authors | |
| Publication Date | June 1, 2022 |
| Published in Issue | Year 2022 Volume: 51 Issue: 3 |