Generalized Woodall Numbers: An Investigation of Properties of Woodall and Cullen Numbers via Their Third Order Linear Recurrence Relations

Year 2022, Volume: 5 Issue: 2, 69 - 81, 30.06.2022

Yüksel Soykan , Vedat İrge

https://doi.org/10.32323/ujma.1057287

Abstract

In this paper, we investigate the generalized Woodall sequences and we deal with, in detail, four special cases, namely, modified Woodall, modified Cullen, Woodall and Cullen sequences. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.

Keywords

Cullen numbers, Tribonacci numbers, Woodall numbers

References

• [1] P. Berrizbeitia, J. G. Fernandes, M. J. Gonz´alez, F. Luca, V. J. M. Huguet, On Cullen numbers which are Both Riesel and Sierpi´nski numbers, Journal of Number Theory, 132 (2012), 2836-2841, http://dx.doi.org/10.1016/j.jnt.2012.05.021
• [2] Y. Bilu, D. Marques, A. Togbe, Generalized Cullen numbers in linear recurrence sequences, Journal of Number Theory, 202 (2019), 412-425, https://doi.org/10.1016/j.jnt.2018.11.025
• [3] I. Bruce, A modified Tribonacci sequence, Fibonacci Quarterly, 22(3) (1984), 244–246.
• [4] M. Catalani, Identities for Tribonacci-related sequences, (2012), arXiv:math/0209179.
• [5] E. Choi, Modular Tribonacci numbers by matrix method, Journal of the Korean Society of Mathematical Education Series B: Pure and Applied. Mathematics, 20(3) (2013), 207–221.
• [6] A. J. C. Cunningham, H. J. Woodall, Factorisation of Q = (2q q) and (q:2q 1), Messenger of Mathematics, 47 (1917), 1–38.
• [7] M. Elia, Derived sequences, The Tribonacci recurrence and cubic forms, Fibonacci Quarterly, 39(2) (2001), 107-115.
• [8] M. C. Er, , Sums of Fibonacci numbers by matrix methods, Fibonacci Quarterly, 22(3) (1984), 204-207.
• [9] J. Grantham, H. Graves, The abc conjecture implies that only finitely many s-Cullen numbers are repunits, (2021) http://arxiv.org/abs/2009.04052v3, MathNT.
• [10] R. Guy, Unsolved Problems in Number Theory (2nd ed.), Springer-Verlag, New York, 1994.
• [11] C. Hooley, Applications of the sieve methods to the theory of numbers, Cambridge University Press, Cambridge, 1976.
• [12] D. Kalman, Generalized Fibonacci numbers by matrix methods, Fibonacci Quarterly, 20(1) (1982), 73-76.
• [13] W. Keller, New Cullen primes, Math. Comput. 64 (1995), 1733-1741.
• [14] P. Y. Lin, De Moivre-Type identities for the Tribonacci numbers, Fibonacci Quarterly, 26 (1988), 131-134.
• [15] F. Luca, P. Stanica, Cullen numbers in binary recurrent sequences, FT. Howard (ed.), Applications of Fibonacci Numbers : Proceedings of the Tenth International Research Conference on Fibonacci Numbers and their Applications, Kluwer Academic Publishers, 9 (2004) 167-175.
• [16] D. Marques, On Generalized Cullen and Woodall numbers that are also Fibonacci numbers, Journal of Integer Sequences, 17 (2014), Article 14.9.4 [17] D. Marques, Fibonacci s-Cullen and s-Woodall numbers, , Journal of Integer Sequences, 18 (2015), Article 15.1.4
• [18] N. K. Meher, S. S. Rout, Cullen numbers in sums of terms of recurrence sequence, (2020), http://arxiv.org/abs/2010.10014v1, MathNT. [19] S. Pethe, Some identities for Tribonacci sequences, Fibonacci Quarterly, 26(2) (1988), 144–151.
• [20] A. Scott, T. Delaney, V. Jr. Hoggatt, The Tribonacci sequence, Fibonacci Quarterly, 15(3) (1977), 193–200.
• [21] A. Shannon, Tribonacci numbers and Pascal’s pyramid, Fibonacci Quarterly, 15(3) (1977), 268-275.
• [22] N.J.A. Sloane, The on-line encyclopedia of integer sequences, http://oeis.org/
• [23] Y. Soykan, Simson Identity of Generalized m-step Fibonacci numbers, Int. J. Adv. Appl. Math. and Mech. 7(2) (2019), 45-56.
• [24] Y. Soykan, Tribonacci and Tribonacci-Lucas sedenions, Mathematics 7(1) (2019), 74.
• [25] Y. Soykan, A study on generalized (r,s,t)-numbers, MathLAB Journal, 7 (2020), 101-129.
• [26] Y. Soykan, On the recurrence properties of getneralized Tribonacci sequence, Earthline Journal of Mathematical Sciences, 6(2) (2021), 253-269, https://doi.org/10.34198/ejms.6221.253269
• [27] W. Spickerman, Binet’s formula for the Tribonacci sequence, Fibonacci Quarterly, 20 (1982), 118–120.
• [28] C. C. Yalavigi, Properties of Tribonacci numbers, Fibonacci Quarterly, 10(3) (1972), 231–246.
• [29] N. Yilmaz, N. Taskara, Tribonacci and Tribonacci-Lucas numbers via the determinants of special matrices, Applied Mathematical Sciences, 8(39) (2014), 1947-1955.