×

A new approach to generalized Fibonacci and Lucas numbers with binomial coefficients. (English) Zbl 1329.11015

Summary: In this study, Fibonacci and Lucas numbers have been obtained by using generalized Fibonacci numbers. In addition, some new properties of generalized Fibonacci numbers with binomial coefficients have been investigated to write generalized Fibonacci sequences in a new direct way. Furthermore, it has been given a new formula for some Lucas numbers.

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Koshy, T., Fibonacci and Lucas Numbers with Applications (2001), John Wiley and Sons Inc.: John Wiley and Sons Inc. NY · Zbl 0984.11010
[2] Stakhov, A., Fibonacci matrices, a generalization of the ‘Cassini formula’, and a new coding theory, Chaos Soliton Fract., 30, 56-66 (2006) · Zbl 1149.94338
[3] Falcoń, S.; Plaza, A., The \(k\)-Fibonacci sequence and the Pascal 2-triangle, Chaos Soliton Fract., 33, 1, 38-49 (2007) · Zbl 1152.11308
[4] Farrokhi, D. G.M., Some remarks on the equation \(F_n = kF_m\) in Fibonacci numbers, J. Integer Seq., 10 (2007)
[5] Spivey, M. Z., Combinatorial sums and finite differences, Discrete Math., 307, 3130-3146 (2007) · Zbl 1129.05006
[7] Akbulak, M.; Bozkurt, D., On the order-\(m\) generalized Fibonacci \(k\)-numbers, Chaos Soliton Fract., 42, 3, 1347-1355 (2009) · Zbl 1198.11012
[8] Demir, A.; Omur, N.; Ulutas, Y. T., Parametrized Fibonacci search method with \(k\)-Lucas numbers, Appl. Math. Comput., 198, 1, 355-360 (2008) · Zbl 1137.65041
[9] Gulec, H. H.; Taskara, N., On the properties of Fibonacci numbers with binomial coefficients, Int. J. Contemp. Math. Sci., 4, 25, 1251-1256 (2009) · Zbl 1196.11030
[10] Benjamin, Arthur T.; Quinn, Jennifer J.; Edward Su, Francis, Phased tilings and generalized Fibonacci identities, Fibonacci Quart., 38, 3, 282-288 (2000) · Zbl 0944.11003
[11] Vajda, S., Fibonacci and Lucas Numbers, and the Golden Section (1989), Theory and applications: Theory and applications John Wiley and Sons, New York · Zbl 0695.10001
[12] Taskara, N.; Uslu, K.; Gulec, H. H., On the properties of Lucas numbers with binomial coefficients, Appl. Math. Lett., 23, 68-72 (2010) · Zbl 1213.11040
[13] El-Mikkawy, M.; Sogabe, T., A new family of \(k\)-Fibonacci numbers, Appl. Math. Comput., 215, 12, 4456-4461 (2010) · Zbl 1193.11012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.