ALTERED NUMBERS OF LUCAS NUMBER SQUARED

Year 2023, Issue: 054, 62 - 75, 30.09.2023

Fikri Köken , Emre Kankal

https://doi.org/10.59313/jsr-a.1251592

Cited By: 1

Abstract

We investigate two types altered Lucas numbers denoted and defined by adding or subtracting a value from the square of the Lucas numbers. We achieve these numbers form as the consecutive products of the Fibonacci numbers. Therefore, consecutive sum-subtraction relations of altered Lucas numbers and their Binet-like formulas are given by using some properties of the Fibonacci numbers. Also, we explore the gcd sequences of r–successive terms of altered Lucas numbers denoted and , , according to the greatest common divisor (gcd) properties of consecutive terms of the Fibonacci numbers. We show that these sequences are periodic or Fibonacci sequences.

Keywords

Altered Lucas numbers, Greatest common divisor (gcd) sequences, Fibonacci sequence, Lucas sequence

References

• [1] OEIS Foundation Inc. (2011). The on-line encyclopedia of integer sequences. http://oeis.org.
• [2] Koshy T. (2001). Fibonacci and Lucas numbers with applications. John Wiley and Sons Inc, NY.
• [3] Chen, K.W. (2011). Greatest common divisors in shifted Fibonacci sequences. Journal of Integer Sequences, 14, 11.4.7.
• [4] Koken, F. (2020). The gcd sequences of the altered Lucas sequences. Annales Mathematicae Silesianae, 34, 2, 222-240. DOI: 10.2478/amsil-2020-0005.
• [5] Cerin, Z. (2013). On factors of sums of consecutive Fibonacci and Lucas numbers. Annales Mathematicae et Informaticae, 41, 19–25.
• [6] Tekcan, A., Gezer, B. and Bizim, O. (2007). Some relations on Lucas numbers and their sums. Advanced Studies in Contemporary Mathematics, 15, 195–211.
• [7] Szalay, L. (2012). Diophantine equations with binary recurrences associated to the Brocard–Ramanujan problem. Portugaliae Mathematica, 69, 3, 213–220.
• [8] Pongsriiam, P. (2017). Fibonacci and Lucas numbers associated with Brocard-Ramanujan equation. Communications of the Korean Mathematical Society, 32, 3, 511-522.
• [9] Kankal E. (2023). The thesis of master of science, Necmettin Erbakan University. The Graduate School of Natural And Applied Science, Konya.