# A Study on the Sums of Squares of Generalized Fibonacci Numbers: Closed Forms of the Sum Formulas Σn k=0 kxkW2 k and Σn k=1 kxkW2- k

## Main Article Content

## Abstract

In this paper, closed forms of the sum formulas Σn k=0 kxkW2 k and Σn k=1 kxkW2 -k for the squares of generalized Fibonacci numbers are presented. As special cases, we give sum formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, 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.

Keywords:

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

## Article Details

How to Cite

*Asian Journal of Advanced Research and Reports*,

*12*(1), 44-67. https://doi.org/10.9734/ajarr/2020/v12i130280

Section

Original Research Article

## References

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Available:https://doi.org/10.9734/ajarr/2020- /v9i130212

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Σn k=0W3 Σ k and n k=1W3

Fibonacci Quarterly; 1965;3(3):161-176.

Horadam AF. A generalized fibonacci sequence. American Mathematical Monthly.

;68:455-459.

Horadam AF. Special properties of the sequence wn(a; b; p; q). Fibonacci Quarterly. 1967;5(5):424-434.

Horadam AF. Generating functions for powers of a certain generalized sequence of numbers. Duke Math. J. 1965;32:437-446.

Sloane NJA. The on-line encyclopedia of integer sequences.

Available: http://oeis.org/

Cˇ erin Z. Formulae for sums of Jacobsthal– Lucas numbers. Int. Math. Forum.

;2(40):1969-1984.

Cˇ erin Z. Sums of squares and products of Jacobsthal numbers. Journal of Integer Sequences. 2007;10.

Gnanam A, Anitha B. Sums of squares Jacobsthal numbers. IOSR Journal of Mathematics. 2015;11(6):62-64.

Kilic¸, E, Tas¸c¸i D. The linear algebra of the pell matrix. Bolet´ın de la Sociedad Matem´ atica Mexicana. 2005;3(11).

Kılıc, E. Sums of the squares of terms of sequence {un}, Proc. Indian Acad. Sci.

(Math. Sci.). 2008;118(1):27-41.

Soykan Y. Closed formulas for the sums of squares of generalized fibonacci numbers. Asian Journal of Advanced Research and Reports. 2020;9(1):23-39.

Available:https://doi.org/10.9734/ajarr/2020- /v9i130212

Soykan Y. Horadam numbers: Sum of the squares of terms of sequence. Int. J. Adv.

Appl. Math. and Mech. In Presss.

Frontczak R. Sums of cubes over odd-index fibonacci numbers. Integers. 2018;18.

Soykan Y. Closed formulas for the sums of cubes of generalized fibonacci numbers: Closed formulas of and

Σn k=0W3 Σ k and n k=1W3