Some addition formulas for Fibonacci, Pell and Jacobsthal numbers
Abstract
In this paper, we obtain a closed form for FΣ_{i=1}^k, PΣ_{i=1}^k and JΣ_{i=1}^k for some positive integers k where Fr, Pr and Jr are the rth Fibonacci, Pell and Jacobsthal numbers, respectively. We also give three open problems for the general cases FΣ_{i=1}^n, PΣ_{i=1}^n and JΣ_{i=1}^n for any arbitrary positive integer n.
Keywords
Fibonacci numbers; Pell numbers; Jacobsthal numbers
References
2. F. Koken and D. Bozkurt, On the Jacobsthal numbers by matrix methods, Int. J. Contemp. Math. Sci. 3 (2008), no. 13, 605–614.
3. T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, New York, 2001.
4. T. Koshy, Pell and Pell-Lucas Numbers with Applications, Springer, New York, 2014.
5. P. Mana, Problem B-152: Fibonacci addition formula, Fibonacci Quart. 7 (1969), no. 3, 336.
6. OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, 2011, http://oeis.org.
Department of Computer Education and Instructional Technologies, Kastamonu University, Turkey Turkey
Department of Mathematics, Institute of Science and Technology, Kastamonu University, Turkey Turkey
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