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.

Keywords

• Altered Fibonacci number
• greatest common divisor (GCD) sequence
• Fibonacci sequence

References

1. T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, New York, 2001.
2. N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences (1964), https://oeis.org/, Accessed 20 Sep 2023.
3. T. Koshy, Elementary Number Theory with Applications, 2nd Edition, Academic Press, California, 2007.
4. U. Dudley, B. Tucker, Greatest Common Divisors in Altered Fibonacci Sequences, Fibonacci Quarterly 9 (1971) 89–91.
5. S. Hernandez, F. Luca, Common Factors of Shifted Fibonacci Numbers, Periodica Mathematica Hungarica 47 (2003) 95–110.
6. 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.
7. Chen, K.W, Greatest Common Divisors in Shifted Fibonacci Sequences, Journal of Integer Sequences 14 (11) (2011) 4–7.
8. F. Koken, The GCD Sequences of the Altered Lucas Sequences, Annales Mathematicae Silesianae 34 (2) (2020) 222–240.

9. N. Robbins, Fibonacci and Lucas numbers of the Forms $w^2-1$, $w^3±1$, Fibonacci Quarterly 19 (4) (1981) 369–373.
10. J. H. E. Cohn, Square Fibonacci Numbers, The Fibonacci Quarterly 2 (2) (1964) 109–113.
11. H. London, R. Finkelstein, On Fibonacci and Lucas Numbers Which are Perfect Powers, The Fibonacci Quarterly 7 (5) (1969) 476–481.
12. R. Finkelstein, On Lucas Numbers Which are One More Than a Square, Fibonacci Quarterly 14 (1) (1973) 340–342.
13. H. C. Williams, On Fibonacci Numbers of the Form $k^2+1$, The Fibonacci Quarterly 13 (2) (1975) 213–214.
14. J. C. Lagarias, D. P. Weisser, Fibonacci and Lucas Cubes, The Fibonacci Quarterly 19 (1) (1981) 39–43.
15. D. Marques, The Fibonacci Version of the Brocard–Ramanujan Diophantine Equation, Portugaliae Mathematica 68 (2) (2011) 185–189.
16. L. Szalay, Diophantine Equations with Binary Recurrences Associated to the Brocard–Ramanujan Problem, Portugaliae Mathematica 69 (3) (2012) 213–220.
17. P. Pongsriiam, Fibonacci and Lucas Numbers Associated with Brocard-Ramanujan Equation, Communications of the Korean Mathematical Society 32 (3) (2017) 511–522.
18. Z. Cerin, On Factors of Sums of Consecutive Fibonacci and Lucas Numbers, Annales Mathematicae et Informaticae 41 (2013) 19–25.
19. 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.
20. F. Koken, E. Kankal, Altered Numbers of Lucas Number Squared, Journal of Scientific Reports A 54 (2023) 62–75.