Korean J. Math. Vol. 24 No. 4 (2016) pp.681-691
DOI: https://doi.org/10.11568/kjm.2016.24.4.681

Some properties of the generalized Fibonacci sequence $\{q_n\}$ by matrix methods

Main Article Content

Sang Pyo Jun
Kwang Ho Choi


In this note, we consider a generalized Fibonacci sequence $\{q_n\}$. We give a generating matrix for $\{q_n\}$. With the aid of this matrix, we derive and re-prove some properties involving terms of this sequence

Article Details


[1] A. Borges, P. Catarino, A. P. Aires, P. Vasco and H. Campos, Two-by-two matrices involving k-Fibonacci and k-Lucas sequences, Applied Mathematical Sciences, 8 (34) (2014), 659–1666. Google Scholar

[2] J. Ercolano, Matrix generators of Pell sequences, Fibonacci Quart., 17 (1) (1979), 71–77. Google Scholar

[3] M. Edson, S. Lewis and O. Yayenie, The k-periodic Fibonacci sequence and extended Binet’s formula, Integer 11 (2011), 1–12. Google Scholar

[4] M. Edson and O. Yayenie, A new generalization of Fibonacci sequence and ex- tended Binet’s formula, Integer 9 (2009), 639–654. Google Scholar

[5] Y. K. Gupta, Y. K. Panwar and O. Sikhwal, Generalized Fibonacci Sequences, Theoretical Mathematics and Applications 2 (2) (2012), 115–124. Google Scholar

[6] Y. K. Gupta, M. Singh and O. Sikhwal, Generalized Fibonacci-Like Sequence Associated with Fibonacci and Lucas Sequences, Turkish Journal of Analysis and Number Theory 2 (6) (2014), 233–238. Google Scholar

[7] A. F. Horadam, A generalized Fibonacci sequences, Amer. Math. Monthly 68 (1961), 455–459. Google Scholar

[8] E. Kilic, Sums of the squares of terms of sequence {un}, Proc. Indian Acad. Sci.(Math. Sci.) 118 (1), February 2008, 27–41. Google Scholar

[9] D. Kalman and R. Mena, The Fibonacci numbers - Exposed, The Mathematical Magazine 2 (2002). Google Scholar

[10] J. R. Silvester, Fibonacci properties by matrix methods, Mathematical Gazette 63 (1979), 188–191. Google Scholar

[11] K. S. Williams, The nth power of a 2 × 2 matrix, Math. Mag. 65 (5) (1992), 336. Google Scholar

[12] O. Yayenie, A note on generalized Fibonacci sequences, Applied Mathematics and Computation 217 (2011), 5603–5611. Google Scholar

[13] H. Zhang and Z. Wu, On the reciprocal sums of the generalized Fibonacci sequences, Adv. Differ. Equ. (2013), Article ID 377 (2013). Google Scholar