Research Article
Altered Numbers of Fibonacci Number Squared
Abstract
We investigate two types of altered Fibonacci numbers obtained by adding or subtracting a specific value $\{a\}$ from the square of the $n^{th}$ Fibonacci numbers $G^{(2)}_{F(n)}(a)$ and $H^{(2)}_{F(n)}(a)$. These numbers are significant as they are related to the consecutive products of the Fibonacci numbers. As a result, we establish consecutive sum-subtraction relations of altered Fibonacci numbers and their Binet-like formulas. Moreover, we explore greatest common divisor (GCD) sequences of r-successive terms of altered Fibonacci numbers represented by $\left\{G^{(2)}_{F(n), r}(a)\right\}$ and $\left\{H^{(2)}_{F(n), r}(a)\right\}$ such that $r\in\{1,2,3\}$ and $a\in\{1,4\}$. The sequences are based on the GCD properties of consecutive terms of the Fibonacci numbers and structured as periodic or Fibonacci sequences.
References
- T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, New York, 2001.
- N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences (1964), https://oeis.org/, Accessed 20 Sep 2023.
- T. Koshy, Elementary Number Theory with Applications, 2nd Edition, Academic Press, California, 2007.
- U. Dudley, B. Tucker, Greatest Common Divisors in Altered Fibonacci Sequences, Fibonacci Quarterly 9 (1971) 89–91.
- S. Hernandez, F. Luca, Common Factors of Shifted Fibonacci Numbers, Periodica Mathematica Hungarica 47 (2003) 95–110.
- J. Spilker, The GCD of the Shifted Fibonacci Sequence, in: J. Sander, J. Steuding, R. Steuding (Eds.), From Arithmetic to Zeta-Functions: Number Theory in Memory of Wolfgang Schwarz, Springer, Cham, 2016, pp. 473–483.
- Chen, K.W, Greatest Common Divisors in Shifted Fibonacci Sequences, Journal of Integer Sequences 14 (11) (2011) 4–7.
- F. Koken, The GCD Sequences of the Altered Lucas Sequences, Annales Mathematicae Silesianae 34 (2) (2020) 222–240.
- N. Robbins, Fibonacci and Lucas numbers of the Forms $w^2-1$, $w^3±1$, Fibonacci Quarterly 19 (4) (1981) 369–373.
- J. H. E. Cohn, Square Fibonacci Numbers, The Fibonacci Quarterly 2 (2) (1964) 109–113.
- H. London, R. Finkelstein, On Fibonacci and Lucas Numbers Which are Perfect Powers, The Fibonacci Quarterly 7 (5) (1969) 476–481.
- R. Finkelstein, On Lucas Numbers Which are One More Than a Square, Fibonacci Quarterly 14 (1) (1973) 340–342.
- H. C. Williams, On Fibonacci Numbers of the Form $k^2+1$, The Fibonacci Quarterly 13 (2) (1975) 213–214.
- J. C. Lagarias, D. P. Weisser, Fibonacci and Lucas Cubes, The Fibonacci Quarterly 19 (1) (1981) 39–43.
- D. Marques, The Fibonacci Version of the Brocard–Ramanujan Diophantine Equation, Portugaliae Mathematica 68 (2) (2011) 185–189.
- L. Szalay, Diophantine Equations with Binary Recurrences Associated to the Brocard–Ramanujan Problem, Portugaliae Mathematica 69 (3) (2012) 213–220.
- P. Pongsriiam, Fibonacci and Lucas Numbers Associated with Brocard-Ramanujan Equation, Communications of the Korean Mathematical Society 32 (3) (2017) 511–522.
- Z. Cerin, On Factors of Sums of Consecutive Fibonacci and Lucas Numbers, Annales Mathematicae et Informaticae 41 (2013) 19–25.
- A. Tekcan, A. Ozkoc, B. Gezer, O. Bizim, Some Relations Involving the Sums of Fibonacci Numbers, Proceedings of the Jangjeon Mathematical Society 11 (1) (2008) 1–12.
- F. Koken, E. Kankal, Altered Numbers of Lucas Number Squared, Journal of Scientific Reports A 54 (2023) 62–75.
Abstract
References
- T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, New York, 2001.
- N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences (1964), https://oeis.org/, Accessed 20 Sep 2023.
- T. Koshy, Elementary Number Theory with Applications, 2nd Edition, Academic Press, California, 2007.
- U. Dudley, B. Tucker, Greatest Common Divisors in Altered Fibonacci Sequences, Fibonacci Quarterly 9 (1971) 89–91.
- S. Hernandez, F. Luca, Common Factors of Shifted Fibonacci Numbers, Periodica Mathematica Hungarica 47 (2003) 95–110.
- J. Spilker, The GCD of the Shifted Fibonacci Sequence, in: J. Sander, J. Steuding, R. Steuding (Eds.), From Arithmetic to Zeta-Functions: Number Theory in Memory of Wolfgang Schwarz, Springer, Cham, 2016, pp. 473–483.
- Chen, K.W, Greatest Common Divisors in Shifted Fibonacci Sequences, Journal of Integer Sequences 14 (11) (2011) 4–7.
- F. Koken, The GCD Sequences of the Altered Lucas Sequences, Annales Mathematicae Silesianae 34 (2) (2020) 222–240.
- N. Robbins, Fibonacci and Lucas numbers of the Forms $w^2-1$, $w^3±1$, Fibonacci Quarterly 19 (4) (1981) 369–373.
- J. H. E. Cohn, Square Fibonacci Numbers, The Fibonacci Quarterly 2 (2) (1964) 109–113.
- H. London, R. Finkelstein, On Fibonacci and Lucas Numbers Which are Perfect Powers, The Fibonacci Quarterly 7 (5) (1969) 476–481.
- R. Finkelstein, On Lucas Numbers Which are One More Than a Square, Fibonacci Quarterly 14 (1) (1973) 340–342.
- H. C. Williams, On Fibonacci Numbers of the Form $k^2+1$, The Fibonacci Quarterly 13 (2) (1975) 213–214.
- J. C. Lagarias, D. P. Weisser, Fibonacci and Lucas Cubes, The Fibonacci Quarterly 19 (1) (1981) 39–43.
- D. Marques, The Fibonacci Version of the Brocard–Ramanujan Diophantine Equation, Portugaliae Mathematica 68 (2) (2011) 185–189.
- L. Szalay, Diophantine Equations with Binary Recurrences Associated to the Brocard–Ramanujan Problem, Portugaliae Mathematica 69 (3) (2012) 213–220.
- P. Pongsriiam, Fibonacci and Lucas Numbers Associated with Brocard-Ramanujan Equation, Communications of the Korean Mathematical Society 32 (3) (2017) 511–522.
- Z. Cerin, On Factors of Sums of Consecutive Fibonacci and Lucas Numbers, Annales Mathematicae et Informaticae 41 (2013) 19–25.
- A. Tekcan, A. Ozkoc, B. Gezer, O. Bizim, Some Relations Involving the Sums of Fibonacci Numbers, Proceedings of the Jangjeon Mathematical Society 11 (1) (2008) 1–12.
- F. Koken, E. Kankal, Altered Numbers of Lucas Number Squared, Journal of Scientific Reports A 54 (2023) 62–75.
There are 20 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Algebra and Number Theory |
| Journal Section | Research Article |
| Authors | |
| Early Pub Date | December 30, 2023 |
| Publication Date | December 31, 2023 |
| Submission Date | September 29, 2023 |
| Published in Issue | Year 2023 Issue: 45 |
