The Hadamard-type Padovan-p Sequences

Year 2020, Volume: 5 Issue: 2, 102 - 109, 31.10.2020

Yeşim Aküzüm

Abstract

In this paper, we define the Hadamard-type Padovan-p sequence by using the Hadamard-type product of characteristic polynomials of the Padovan sequence and the Padovan-p sequence. Also, we derive the generating matrices for these sequences. Then using the roots of characteristic polynomial of the Hadamard-type Padovan-p sequence, we produce the Binet formula for the Hadamard-type Padovan-p numbers. Also, we give the permanental, determinantal, combinatorial, exponential representations and the sums of the Hadamard-type Padovan-p numbers.

Keywords

The Hadamard product, The Padovan-p sequence, Matrix, Representation

References

• [1] Akuzum Y, Deveci O. The Hadamard-type k-step Fibonacci sequences. is submitted.
• [2] Akuzum Y, Deveci O. Sylvester-Padovan-Jacobsthal-Type sequences. Maejo Int. J. Sci. Technol. 11(03), 2017, 236-248.
• [3] Brualdi RA, Gibson PM. Convex polyhedra of doubly stochastic matrices I: applications of permanent function. J. Combin. Theory. 22, 1997, 194-230.
• [4] Chen WYC, Louck JC. The combinatorial power of the companion matrix. Linear Algebra Appl. 232, 1996, 261-278.
• [5] Deveci O. The Padovan-Circulant sequences and their applications. Math. Reports. 20(70), 2018, 401-416.
• [6] Deveci O, Akuzum Y, Karaduman E. The Pell-Padovan p-sequences and its applications. Util. Math. 98, 2015, 327-347.
• [7] Deveci O, Karaduman E. On the Padovan p-numbers. Hacettepe J. Math. Stat. 46(4), 2017, 579-592.
• [8] Erdag O, Deveci O. The arrowhead-Fibonacci-Random-type sequences. Util. Math. 113, 2019, 271-287.
• [9] Erdag O, Shannon AG, Deveci O. The arrowhead-Pell-Random-type sequences. Notes Number Theory Disc. Math. 24(1), 2018, 109-119.
• [10] Frey DD, Sellers JA. Jacobsthal numbers and alternating sign matrices. J. Integer Seq. 3, Article 00.2.3., 2000.
• [11] Gogin ND, Myllari AA. The Fibonacci-Padovan sequence and MacWilliams transform matrices. Programing and Computer Software, published in Programmirovanie. 33(2), 2007, 74-79.
• [12] Kalman D. Generalized Fibonacci numbers by matrix methods. Fibonacci Quart. 20(1), 1982, 73-76.
• [13] Kilic E, Tasci D. The Generalized Binet Formula, Representation and Sums of The Generalized Order-k Pell Numbers. Taiwan. J. Math. 10(6), 2006, 1661-1670.
• [14] Ozkan E. On truncated Fibonacci sequences. Indian J. Pure Appl. Math. 38(4), 2007, 241-251.
• [15] Ozkan E, Altun I, Gocer AA. On relationship among a new family of k-Fibonacci, k-Lucas numbers, Fibonacci and Lucas numbers. Chiang Mai J. Sci. 44, 2017, 1744-1750.
• [16] Shannon AG, Anderson PG, Horadam AF. Properties of cordonnier Perrin and Van der Laan numbers. Internat. J. Math. Ed. Sci. Tech. 37(7), 2006, 825-831.
• [17] Shannon AG, Bernstein L. The Jacobi-Perron algorithm and the algebra of recursive sequences. Bull. Australian Math. Soc. 8, 1973, 261-277.
• [18] Shannon AG, Deveci O, Erdag O. Generalized Fibonacci numbers and Bernoulli polynomials. Notes Number Theory Disc. Math. 25, 2019, 193-19.
• [19] Tascı D., Firengiz MC. Incomplete Fibonacci and Lucas p numbers. Math. Comput. Modelling. 52(9), 2010, 1763-1770.
• [20] Wolfram Research: Inc. Mathematica, Version 10.0: Champaign, Illinois. 2014.