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 $\{ w_{n}\}$.
📌 3 Maddede Bu Makale
🎯
Amaç & Hipotez: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 $\{ w_{n}\}$.
📊
Temel Bulgular & Veri: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 $\{ w_{n}\}$.
💡
Sonuç & Katkı: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 $\{ w_{n}\}$.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Horadam sequence · generalized Fibonacci sequence · generalized Lucas 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 $\{ w_{n}\}$.
Tan, E., Leung, H. H. (2020). A note on congruence properties of the generalized bi-periodic Horadam sequence. Hacettepe Journal of Mathematics and Statistics, Volume: 49 Issue: 6. https://doi.org/10.15672/hujms.539587
G. Bilgici, Two generalizations of Lucas sequence, Appl. Math. Comput. 245, 526– 538, 2014
L. Carlitz and H.H. Ferns, Some Fibonacci and Lucas Identities, Fibonacci Quart. 8 (1), 61–73, 1970.
M. Edson and O. Yayenie, A new generalizations of Fibonacci sequences and extended Binet’s Formula, Integers, 9, 639–654, 2009.
A.F. Horadam, Basic Properties of a Certain Generalized Sequence of Numbers, Fibonacci Quart. 3 (3), 161–76, 1965.
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.
E. Kilic and E. Tan, More General Identities Involving The Terms Of $\{Wn(a,b;p,q)\}$, Ars Comb. 93, 459–461, 2009.
D. Panario, M. Sahin and Q. Wang, A family of Fibonacci-like conditional sequences, Integers, 13 (A78), 2013.
I.D. Ruggles, Some Fibonacci results using Fibonacci-type sequences, Fibonacci Quart. 1 (2), 75–80, 1963.
M. Sahin, The Gelin-Cesaro identity in some conditional sequences, Hacet. J. Math. Stat. 40 (6), 855–861, 2011.
Z. Siar and R. Keskin, Some new identities concerning generalized Fibonacci and Lucas numbers, Hacet. J. Math. Stat. 42 (3), 211-222, 2013.
E. Tan, On bi-periodic Fibonacci and Lucas numbers by matrix method, Ars Combin. 133, 107–113, 2017.
E. Tan, Some properties of the bi-periodic Horadam sequences, Notes Number Theory Discrete Math. 23 (4), 56–65, 2017.
E. Tan and A.B. Ekin, Bi-Periodic Incomplete Lucas Sequences, Ars Combin. 123, 371–380, 2015.
E. Tan and A.B. Ekin, Some Identities On Conditional Sequences By Using Matrix Method, Miskolc Math. Notes, 18 (1), 469–477, 2017.
E. Tan and H.H. Leung, Some basic properties of the generalized bi-periodic Fibonacci and Lucas sequences, Adv. Difference Equ. 2020 (26), 2020.
O. Yayenie, A note on generalized Fibonacci sequence, Appl. Math. Comput. 217, 5603–5611, 2011.
J. Yang and Z. Zhang, Some identities of the generalized Fibonacci and Lucas sequences, Appl. Math. Comput. 339, 451–458, 2018.
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.
Z. Zhang and M. Liu, Generalizations of some identities involving generalized second-order integer sequences, Fibonacci Quart. 36 (4), 327–328, 1998.