Generalized Fibonacci Numbers: Sum Formulas of the Squares of Terms

Authors

  • Yüksel Soykan Zonguldak Bülent Ecevit University, 67100, Zonguldak, Turkey

Keywords:

Fibonacci numbers, Lucas numbers, Pell numbers, Jacobsthal numbers, sum formulas

Abstract

In this paper, closed forms of the sum formulas nk=1kWk2 and nk=1kW2−k for the squares of generalized Fibonacci numbers are presented. As special cases, we give summation formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas numbers. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery. Our work generalize second order recurrence relations.

Downloads

Download data is not yet available.

Author Biography

Yüksel Soykan, Zonguldak Bülent Ecevit University, 67100, Zonguldak, Turkey

Department of Mathematics, Art and Science Faculty

References

Čerin, Z., Formulae for sums of Jacobsthal–Lucas numbers, Int. Math. Forum, 2(40), 1969–1984, 2007. https://doi.org/10.12988/imf.2007.07178.

Čerin, Z., Sums of Squares and Products of Jacobsthal Numbers. Journal of Integer Sequences, 10, Article 07.2.5,2007.

Chen, L., Wang, X., The Power Sums Involving Fibonacci Polynomials and Their Applications, Symmetry, 11,2019, doi.org/10.3390/sym11050635.

Frontczak, R., Sums of powers of Fibonacci and Lucas numbers: A new bottom-up approach, Notes on NumberTheory and Discrete Mathematics, 24(2), 94–103, 2018. DOI: 10.7546/nntdm.2018.24.2.94-103.

Frontczak, R., Sums of Cubes Over Odd-Index Fibonacci Numbers, Integers, 18, 2018.

Gnanam, A., Anitha, B., Sums of Squares Jacobsthal Numbers. IOSR Journal of Mathematics, 11(6), 62-64. 2015. DOI: 10.9790/5728-11646264

Horadam, A.F., A Generalized Fibonacci Sequence, American Mathematical Monthly, Vol. 68, pp. 455-459, 1961. https://doi.org/10.1080/00029890.1961.11989696.

Horadam, A. F., Basic Properties of a Certain Generalized Sequence of Numbers, Fibonacci Quarterly 3.3, 161-176,1965.

Horadam, A. F., Special Properties of The Sequence wn(a,b;p,q), Fibonacci Quarterly, Vol. 5, No. 5, pp. 424-434,1967.

Horadam, A. F., Generating functions for powers of a certain generalized sequence of numbers. Duke Math. J 32, 437-446, 1965. https://doi.org/10.1215/S0012-7094-65-03244-8.

Kiliç, E., Taşçi, D., The Linear Algebra of The Pell Matrix, Boletín de la Sociedad Matemática Mexicana, 3(11),2005.

Kılıc, E., Sums of the squares of terms of sequence{un}, Proc. Indian Acad. Sci. (Math. Sci.) 118(1), 27–41, 2008. https://doi.org/10.1007/s12044-008-0003-y.

Prodinger, H., Sums of Powers of Fibonacci Polynomials, Proc. Indian Acad. Sci. (Math. Sci.), 119(5), 567-570,2009. https://doi.org/10.1007/s12044-009-0060-x.

Prodinger, H., Selkirk, S.J., Sums of Squares of Tetranacci Numbers: A Generating Function Approach, 2019, http://arxiv.org/abs/1906.08336v1.

Raza, Z., Riaz, M., Ali, M.A., Some Inequalities on the Norms of Special Matrices with Generalized Tribonacciand Generalized Pell-Padovan Sequences, arXiv, 2015, http://arxiv.org/abs/1407.1369v2

Schumacher, R., How to sum the squares of the Tetranacci numbers and the Fibonacci m-step numbers. Fibonacci Quarterly, 57:168–175, 2019.

Sloane, N.J.A., The on-line encyclopedia of integer sequences. Available: http://oeis.org/

Soykan, Y. Closed Formulas for the Sums of Squares of Generalized Fibonacci Numbers, Asian Journal of AdvancedResearch and Reports, 9(1), 23-39, 2020. https://doi.org/10.9734/ajarr/2020/v9i130212

Wamiliana., Suharsono., Kristanto, P. E., Counting the sum of cubes for Lucas and Fibonacci Numbers, Science and Technology Indonesia, 4(2), 31-35, 2019. https://doi.org/10.26554/sti.2019.4.2.31-35.

Published

2020-04-30

How to Cite

Yüksel Soykan. (2020). Generalized Fibonacci Numbers: Sum Formulas of the Squares of Terms. MathLAB Journal, 5, 46-62. Retrieved from http://purkh.com/index.php/mathlab/article/view/732

Issue

Section

Research Articles