The Representation and Finite Sums of the Padovan-p Jacobsthal Numbers

Abstract

In this paper, we consider the Padovan-p Jacobsthal sequence and then we discuss the connection of the Padovan-p Jacobsthal numbers and Jacobsthal numbers. Furthermore, we give the permanental, determinantal, combinatorial, and exponential representations, and the sums of the Padovan-p Jacobsthal numbers by the aid of the generating function and generating matrix of this sequence.

Keywords

• Padovan-p Jacobsthal number
• Matrix
• Representation
• Sum

Kaynakça

1. Akuzum Y. The Hadamard-type Padovan-p Sequences, Turkish Journal of Science. 5(2), 2020, 102−109.
2. Akuzum Y. The Padovan-p Jacobsthal Numbers and Binet Formulas. 3. International Baku Scientific Research Congress. 15-16 October 2021, Baku Eurasıa University, Baku, Azerbaijan.
3. Akuzum Y, Deveci O, Shannon AG. On the Pell p-circulant Sequences. Notes on Number Theory and Discrete Mathematics. 23(2), 2017, 91−103.
4. Akuzum Y, Deveci O. The Arrowhead-Jacobsthal Sequences. Mathematica Montisnigri. Vol-LI, 2021, 31−44.
5. Bradie B. Extension and refinements of some properties of sums involving Pell number. Missouri Journal of Mathematical Sciences. 22(1), 2010, 37−43.
6. Brualdi RA, Gibson PM. Convex polyhedra of doubly stochastic matrices I: applications of permanent function. Journal of Combinatorial Theory, Series A. 22(2), 1977, 194−230.
7. Chen WYC, Louck JD. The combinatorial power of the companion matrix. Linear Algebra and its Applications. 232, 1996, 261−278.
8. Deveci O. On the connections among Fibonacci, Pell, Jacobsthal and Padovan numbers. Notes on Number Theory and Discrete Mathematics. 27(2), 2021, 111−128.

9. Deveci O, Karaduman E. On the Padovan p-numbers. Hacettepe Journal of Mathematics and Statistics. 46(4), 2017, 579−592.
10. Deveci O, Adiguzel Z, Akuzum Y. On the Jacobsthal-circulant-Hurwitz Numbers. Maejo International Journal of Science and Technology. 14(1), 2020, 56−67.
11. Erdag O, Deveci O. On The Connections Between Padovan Numbers and Padovan p-Numbers. International Journal of Open Problems in Computer Science and Mathematics. 13(4), 2020, 33−47.
12. Horadam A. Jacobsthal representations numbers. Fibonacci Quarterly. 34(1), 1996, 40−54.
13. Horadam A. Applications of modified Pell numbers to representations. Ulam Quarterly. 3(1), 1994, 34−53.
14. Kalman D. Generalized Fibonacci numbers by matrix methods. Fibonacci Quarterly. 20(1), 1982, 73−76.
15. Kilic E. The Binet formula, sums and representations of generalized Fibonacci p-numbers. European Journal of Combinatorics. 29(3), 2008, 701−711.
16. Kilic E. The generalized Pell (p, i)-numbers and their Binet formulas, combinatorial representations, sums. Chaos, Solitons and Fractals. 40(4), 2009, 2047−2063.
17. Kilic E, Tasci D. The generalized Binet formula, representation and sums of the generalized order-k Pell numbers. Taiwanese Journal of Mathematics. 10(6), 2006, 1661−1670.
18. Koken F, Bozkurt D. On the Jacobsthal numbers by matrix methods. International Journal of Contemporary Mathematical Sciences. 3(13), 2008, 605−614.
19. Lancaster P, Tismenetsky M. The theory of matrices: with applications. Elsevier. 1985.
20. Lidl R, Niederreiter H. Introduction to finite fields and their applications. Cambridge UP. 1986.
21. Shannon AG, Erdag O, Deveci O. On the Connections Between Pell Numbers and Fibonacci p-Numbers. Notes on Number Theory and Discrete Mathematics. 27(1), 2021, 148−160.