Open AccessGeneralized Fibonacci Numbers with Indices in Arithmetic Progression and Sum of Their Squares: The Sum Formula ∑nk=0 xkW2mk+j
Author(s) -
Yüksel Soykan
Publication year - 2021
Publication title -
journal of advances in mathematics and computer science
Language(s) - English
Resource type - Journals
ISSN - 2456-9968
DOI - 10.9734/jamcs/2021/v36i630371
Subject(s) - fibonacci number , lucas number , mathematics , pisano period , mathematical proof , fibonacci polynomials , combinatorics , lucas sequence , arithmetic , mathematical induction , discrete mathematics , algebra over a field , pure mathematics , classical orthogonal polynomials , orthogonal polynomials , geometry
In this paper, closed forms of the sum formulas ∑n k=0 xkWmk 2 +j for 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.