ON THE BINOMIAL SUMS OF HORADAM SEQUENCE

Abstract

The main purpose of this paper is to establish some new properties of Horadam numbers in terms of binomial sums. By that, we can obtain these special numbers in a new and direct way. Moreover, some connections between Horadam and generalized Lucas numbers are revealed to get a more strong result.

Keywords

• Binomial sums,Horadam numbers,generalized Lucas numbers

References

1. [1] A.F. Horadam, Basic properties of a certain generalized sequence of numbers, The Fibonacci Quarterly 3, (1965), 161-176.
2. [2] E. Kilic, E. Tan, On Binomial Sums for the General Second Order Linear Recurrence, Integers 10 (2010), 801{806.
3. [3] E. Kilic, Y. Turker Ulutas, N. Omur, Sums of Products of the Terms of Generalized Lucas Sequence fVkng ; Hacettepe Journal of Mathematics and Statistics, Volume 40(2), (2011), 147 {161.
4. [4] G. Udrea, A note on sequence of A.F. Horadam, Portugaliae Mathematica 53(24), (1996), 143-144.
5. [5] H.H. Gulec, N. Taskara, On The Properties of Fibonacci Numbers with Binomial Coecients, Int. J. of Contemp. Math. Sci. 4(25), (2009), 1251-1256.
6. [6] MS. El Naschie, The Fibonacci code behind super strings and P-Branes, an answer to M. Kakus fundamental question, Chaos, Solitons & Fractals 31(3), (2007), 537-47.
7. [7] N. Taskara, K. Uslu, H.H. Gulec, On the properties of Lucas numbers with binomial coe- cients, Appl. Math. Lett. 23(1), (2010), 68-72.
8. [8] N. Taskara, K. Uslu, Y. Yazlik, N. Yilmaz, The Construction of Horadam Numbers in Terms of the Determinant of Tridiagonal Matrices, Numerical Analysis and Applied Mathematics, AIP Conference Proceedings 1389, (2011), 367-370.

9. [9] N. Yilmaz, N. Taskara, K. Uslu & Y. Yazlik, On The Binomial Sums of k-Fibonacci and k- Lucas Sequences, Numerical Analysis and Applied Mathematics, AIP Conference Proceedings 1389, (2011), 341-344.
10. [10] S. Falcon, On the k-Lucas Numbers, Int. J. Contemp. Math. Sciences, Vol. 6, no. 21, (2011), 1039-1050.
11. [11] S. Falcon, A. Plaza, On k-Fibonacci numbers of arithmetic indexes, Applied Mathematics and Computation 208, (2009), 180-185.
12. [12] S. Falcon, A. Plaza, the k-Fibonacci sequence and the Pascal 2-triangle, Chaos, Solitons & Fractals 33, (2007), 38-49.
13. [13] S. Vajda, Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications, Dover Publications New York (2007).
14. [14] T. Horzum, E.G. Kocer, On Some Properties of Horadam Polynomials, Int. Math. Forum, 4, 25, (2009), 1243-1252.
15. [15] T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons Inc, NY (2001).
16. [16] T. Mansour, A formula for the generating functions of powers of Horadam's sequence, Aus- tralasian Journal of Combinatorics 30, (2004), 207-212.