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
Y¨uksel Soykan *
Department of Mathematics, Faculty of Art and Science, Zonguldak B¨ ulent Ecevit University, 67100, Zonguldak, Turkey.
*Author to whom correspondence should be addressed.
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.