On the Padovan p-numbers

Year 2017, Volume: 46 Issue: 4, 579 - 592, 01.08.2017

Ömür Deveci , Erdal Karaduman

Abstract

In this paper, we define the Padovan p-numbers and then we obtain their miscellaneous properties such as the generating matrix, the Binet
formula, the generating function, the exponential representation, the combinatorial representations, the sums and permanental representation. Also, we study the Padovan p-numbers modulo m. Furthermore, we define Padovan p-orbit of a finite group and then, we obtain the length of the Padovan p-orbits of the quaternion group $Q_{2^n}$, $(n\geq 3)$.

Keywords

The Padovan p-numbers, Matrix, Sequence, Group, Length

References

• Aydin H. and Dikici R., General Fibonacci sequences in finite groups, Fibonacci Quart., 36 (3), 216-221, 1998.
• Brualdi R. A. and Gibson P. M., Convex polyhedra of doubly stochastic matrices I: applica- tions of permanent function, J. Combin. Theory, 22, 194-230, 1977.
• Campbell C. M., Doostie H. and Robertson E. F., Fibonacci length of generating pairs in groups in Applications of Fibonacci Numbers, Vol. 3 Eds. G. E. Bergum et al. Kluwer Academic Publishers, 27-35, 1990.
• Chen W. Y. C. and Louck J. D., The combinatorial power of the companion matrix, Linear Algebra Appl., 232, 261-278, 1996.
• Deveci O., The polytopic-k-step Fibonacci sequences in finite groups, Discrete Dyn. Nat. Soc., 431840-1-431840-13, 2011.
• Deveci O. and Karaduman E., The Pell sequences in finite groups, Util. Math., 96, 263-276, 2015.
• Deveci O., The Pell-Padovan sequences and the Jacobsthal-Padovan sequences in nite groups, Util. Math., 98, 257-270, 2015.
• Doostie H. and Hashemi M., Fibonacci lengths involving the Wall number k(n), J. Appl. Math. Comput., 20 (1-2), 171-180, 2006.
• Falcon S. and Plaza A., k-Fibonacci sequences modulo m, Chaos Solitons Fractals, 41 (1), 497-504, 2009.
• Frey D. D. and Sellers J. A., Jacobsthal numbers and alternating sign matrices, J. Integer Seq., 3, Article 00.2.3, 2000.
• Gil J. B., Wiener M. D. and Zara C., Complete Padovan sequences in nite elds, http://arxiv.org/pdf/math/0605348.pdf.
• Gogin N. D. and Myllari A. A., The Fibonacci-Padovan sequence and MacWilliams transform matrices, Programing and Computer Software, published in Programmirovanie, 33(2), 74-79, 2007.
• http://mathworld.wolfram.com/Padovan Sequence.html
• Kalman D., Generalized Fibonacci numbers by matrix methods, Fibonacci Quart., 20 (1), 73-76, 1982.
• Kilic E., The generalized Pell (p,i)-numbers and their Binet formulas, combinatorial repre- sentations, sums, Chaos Solitons Fractals, 40 (4), 2047-2063, 2009.
• Kilic E. and Tasci D., The generalized Binet formula, representation and sums of the generalized order-k Pell numbers, Taiwanese J. Math., 10 (6), 1661-1670, 2006.
• Knox S. W., Fibonacci sequences in finite groups, Fibonacci Quart., 30 (2), 116-120, 1992.
• Knuth D. E. and Bendix P. B, Simple word problems in universal algebra, In: Computational problems in abstract algebra, (edited by L. Leech), Pergamon Press, Oxford, 263-297, 1970.