The GCD Sequences of the Altered Lucas Sequences

Fikri Koken 1
  • 1 Eregli Kemal Akman Vocational School, Necmettin Erbakan University, Konya, Turkey


In this study, we give two sequences {L+n}n≥1 and {Ln}n≥1 derived by altering the Lucas numbers with {±1, ±3}, terms of which are called as altered Lucas numbers. We give relations connected with the Fibonacci Fn and Lucas Ln numbers, and construct recurrence relations and Binet’s like formulas of the L+n and Ln numbers. It is seen that the altered Lucas numbers have two distinct factors from the Fibonacci and Lucas sequences. Thus, we work out the greatest common divisor (GCD) of r-consecutive altered Lucas numbers. We obtain r-consecutive GCD sequences according to the altered Lucas numbers, and show that their GCD sequences are unbounded or periodic in terms of values r.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] K.-W. Chen, Greatest common divisors in shifted Fibonacci sequences, J. Integer. Seq. 14 (2011), no. 4, Article 11.4.7, 8 pp.

  • [2] U. Dudley and B. Tucker, Greatest common divisors in altered Fibonacci sequences, Fibonacci Quart. 9 (1971), no. 1, 89–91.

  • [3] L. Hajdu and M. Szikszai, On the GCD-s of k consecutive terms of Lucas sequences, J. Number Theory 132 (2012), no. 12, 3056–3069.

  • [4] S. Hernández and F. Luca, Common factors of shifted Fibonacci numbers, Period. Math. Hungar. 47 (2003), no. 1-2, 95–110.

  • [5] L. Jones, Primefree shifted Lucas sequences, Acta Arith. 170 (2015), no. 3, 287–298.

  • [6] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, New York, 2001.

  • [7] A. Rahn and M. Kreh, Greatest common divisors of shifted Fibonacci sequences revisited, J. Integer. Seq. 21 (2018), no. 6, Art. 18.6.7, 12 pp.

  • [8] J. Spilker, The GCD of the shifted Fibonacci sequence, in: J. Sander et al. (eds.), From Arithmetic to Zeta-Functions, Springer, Cham, 2016, pp. 473–483.


Journal + Issues