2013

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

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,

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[1] G. Bilgici, Two generalizations of Lucas sequence, Appl. Math. Comput. 245, 526– 538, 2014
[2] L. Carlitz and H.H. Ferns, Some Fibonacci and Lucas Identities, Fibonacci Quart. 8 (1), 61–73, 1970.
[3] M. Edson and O. Yayenie, A new generalizations of Fibonacci sequences and extended Binet’s Formula, Integers, 9, 639–654, 2009.
[4] A.F. Horadam, Basic Properties of a Certain Generalized Sequence of Numbers, Fibonacci Quart. 3 (3), 161–76, 1965.
[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] E. Kilic and E. Tan, More General Identities Involving The Terms Of $\{Wn(a,b;p,q)\}$, Ars Comb. 93, 459–461, 2009.
[7] D. Panario, M. Sahin and Q. Wang, A family of Fibonacci-like conditional sequences, Integers, 13 (A78), 2013.
[8] I.D. Ruggles, Some Fibonacci results using Fibonacci-type sequences, Fibonacci Quart. 1 (2), 75–80, 1963.
[9] M. Sahin, The Gelin-Cesaro identity in some conditional sequences, Hacet. J. Math. Stat. 40 (6), 855–861, 2011.
[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] E. Tan, On bi-periodic Fibonacci and Lucas numbers by matrix method, Ars Combin. 133, 107–113, 2017.
[12] E. Tan, Some properties of the bi-periodic Horadam sequences, Notes Number Theory Discrete Math. 23 (4), 56–65, 2017.
[13] E. Tan and A.B. Ekin, Bi-Periodic Incomplete Lucas Sequences, Ars Combin. 123, 371–380, 2015.
[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] 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] O. Yayenie, A note on generalized Fibonacci sequence, Appl. Math. Comput. 217, 5603–5611, 2011.
[17] J. Yang and Z. Zhang, Some identities of the generalized Fibonacci and Lucas sequences, Appl. Math. Comput. 339, 451–458, 2018.
[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] Z. Zhang and M. Liu, Generalizations of some identities involving generalized second-order integer sequences, Fibonacci Quart. 36 (4), 327–328, 1998.

Hacettepe Journal of Mathematics and Statistics-Cover

Yayın Aralığı: Yılda 6 Sayı
Başlangıç: 2002
Yayıncı: Hacettepe Üniversitesi Fen Fakultesi

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