A note on congruence properties of the generalized bi-periodic Horadam sequence

Abstract

In this paper, we consider a generalization of Horadam sequence $\left\{w_{n}\right\} $ which is defined by the recurrence $w_{n}=aw_{n-1}+cw_{n-2},$ if $n$ is even, $w_{n}=bw_{n-1}+cw_{n-2},$ if $n$ is odd with arbitrary initial conditions $w_{0},w_{1}$ and nonzero real numbers $a,b,$ and $c.$ We investigate some congruence properties of the generalized Horadam sequence $\{ w_{n}\}$.

Keywords

• Horadam sequence
• generalized Fibonacci sequence
• generalized Lucas sequence
• congruence

References

1. [1] G. Bilgici, Two generalizations of Lucas sequence, Appl. Math. Comput. 245, 526– 538, 2014
2. [2] L. Carlitz and H.H. Ferns, Some Fibonacci and Lucas Identities, Fibonacci Quart. 8 (1), 61–73, 1970.
3. [3] M. Edson and O. Yayenie, A new generalizations of Fibonacci sequences and extended Binet’s Formula, Integers, 9, 639–654, 2009.
4. [4] A.F. Horadam, Basic Properties of a Certain Generalized Sequence of Numbers, Fibonacci Quart. 3 (3), 161–76, 1965.
5. [5] L.C. Hsu and M.S. Jiang, A kind of invertible graphical process for finding reciprocal formulas with applications, Acta Sci. Nat. Univ. Jilinensis, 4, 43–55, 1980.
6. [6] E. Kilic and E. Tan, More General Identities Involving The Terms Of $\{Wn(a,b;p,q)\}$, Ars Comb. 93, 459–461, 2009.
7. [7] D. Panario, M. Sahin and Q. Wang, A family of Fibonacci-like conditional sequences, Integers, 13 (A78), 2013.
8. [8] I.D. Ruggles, Some Fibonacci results using Fibonacci-type sequences, Fibonacci Quart. 1 (2), 75–80, 1963.

9. [9] M. Sahin, The Gelin-Cesaro identity in some conditional sequences, Hacet. J. Math. Stat. 40 (6), 855–861, 2011.
10. [10] Z. Siar and R. Keskin, Some new identities concerning generalized Fibonacci and Lucas numbers, Hacet. J. Math. Stat. 42 (3), 211-222, 2013.
11. [11] E. Tan, On bi-periodic Fibonacci and Lucas numbers by matrix method, Ars Combin. 133, 107–113, 2017.
12. [12] E. Tan, Some properties of the bi-periodic Horadam sequences, Notes Number Theory Discrete Math. 23 (4), 56–65, 2017.
13. [13] E. Tan and A.B. Ekin, Bi-Periodic Incomplete Lucas Sequences, Ars Combin. 123, 371–380, 2015.
14. [14] E. Tan and A.B. Ekin, Some Identities On Conditional Sequences By Using Matrix Method, Miskolc Math. Notes, 18 (1), 469–477, 2017.
15. [15] E. Tan and H.H. Leung, Some basic properties of the generalized bi-periodic Fibonacci and Lucas sequences, Adv. Difference Equ. 2020 (26), 2020.
16. [16] O. Yayenie, A note on generalized Fibonacci sequence, Appl. Math. Comput. 217, 5603–5611, 2011.
17. [17] J. Yang and Z. Zhang, Some identities of the generalized Fibonacci and Lucas sequences, Appl. Math. Comput. 339, 451–458, 2018.
18. [18] Z. Zhang, Some properties of the generalized Fibonacci sequences $c_{n}=c_{n-1}+c_{n-2}+r$, Fibonacci Quart. 35 (2), 169–171, 1997.
19. [19] Z. Zhang and M. Liu, Generalizations of some identities involving generalized second-order integer sequences, Fibonacci Quart. 36 (4), 327–328, 1998.