Generalized Fibonacci Numbers with Indices in Arithmetic Progression and Sum of Their Squares: The Sum Formula ∑nk=0 xkW2mk+j
Journal of Advances in Mathematics and Computer Science · pp. 30–62 · Published 29 Jul 2021
10.9734/jamcs/2021/v36i630371Abstract
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.
References (10)
- 1 A Generalized Fibonacci Sequence [DOI]
- 2 The on-line encyclopedia of integer sequences [DOI]
- 3 Pell and Pell–Lucas Numbers with Applications [DOI]
- 4 THE GENERALIZED BINET FORMULA, REPRESENTATION AND SUMS OF THE GENERALIZED ORDER-$k$ PELL NUMBERS [DOI]
- 5 On The Jacobsthal Numbers By Matrix Method
- 6 Formulae for sums of Jacobsthal-Lucas numbers [DOI]
- 7 Sums of Squares and Products of Jacobsthal Numbers
- 8 On Generalized Third-Order Pell Numbers [DOI]
- 9 On generalizations of the Jacobsthal sequence [DOI]
- 10 A Study of Generalized Fourth-Order Pell Sequences [DOI]
Cited by 2
Yüksel Soykan, Erkan Taşdemir, Can Murat Dikmen · Journal of Innovative Applied Mathematics and Computational Sciences · 2021
Showing 1 of 2 known citations — external sources report more than can currently be individually listed.
Related research
- Closed Formulas for the Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of Σn k=0W3 Σ k and n k=1W3 — shares topic coverage
- Some Properties of Generalized Fibonacci Numbers: Identities, Recurrence Properties and Closed Forms of the Sum Formulas ∑nk=0 xkWmk+j — shares topic coverage
- On Summing Formulas for Horadam Numbers — shares topic coverage
- Closed Formulas for the Sums of Squares of Generalized Fibonacci Numbers — shares topic coverage
- 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 — shares topic coverage
Article metrics
Real usage data collected on this platform.
30
Page views
0
PDF downloads
4
Outbound clicks
2
Citations
Views over time
Views by country
Approximate, from request IP at view time — not citizenship or institution. Countries with fewer than 5 views are grouped as "Other".
Traffic sources
Referring site, by host.
Views and downloads exclude known bots/crawlers. Citations combines this platform's own DOI-resolved index with each external source's own reported total — see Cited by above for individually listed citing works. Last refreshed 0 seconds ago.