On The Generalizations of Petranacci Sequences

Year 2024, Volume: 12 Issue: 2, 172 - 178, 28.10.2024

K. L. Verma

Abstract

In this article, a generalizations of the Pentanacci sequence, which is generated by the fifth-order recurrence relation ${{V}_{n}}\left( {{a}_{j}},{{p}_{j}} \right)=\sum\limits_{j=1}^{5}{{{p}_{j}}{{V}_{n-j}}}$, $n>5$, with the initial terms $ {{V}_{j}}={{a}_{j}},$ where ${{a}_{j}},{{p}_{j}},\text{ }j=1,2,3,4,5$ are any non–zero real numbers is studied. Generating function and Binet’s formula are established for this sequence in the denotative form. Noted sequences generated by the recurrence relations of lower orders are contained implicitly in this generalization and are discussed as special cases. A graphical representation is presented to exhibit the relations how the terms of these sequences are related and varies with different ${{a}_{j}},{{p}_{j}}.$ Pentanacci constant are also studied and represented in the tabular form, it is shown that it depends on the coefficients of the recurrence relations only and has no effect of the initial terms.

Keywords

Pentanacci sequence, Generating function, Binet formula, Pentanacci constant, Pentanacci Identity

Ethical Statement

It is declared that during the preparation process of this study, scientific and ethical principles were followed .

Supporting Institution

None

Project Number

None

Thanks

KL Verma

References

• [1] T. Sarisahin and A. Nalli, On the Pentanacci numbers. Mathematical and Computational Applicationss Vol 10 (3), (2014), 255-262.
• [2] G. P Dresden and Z. Du, A Simplified Binet Formula for k-Generalized Fibonacci Numbers, Journal of Integer Sequences,Vol. 17 (2014), Article 14.4.7.
• [3] L. R.,Natividad, On Solving Fibonacci-Like Sequences of Fourth, Fifth and Sixth Order, Int. J. of Mathematics and Computing, Vol.3 (2), (2013), 38-40.
• [4] Yuksel Soykan, On A Generalized Pentanacci Sequence”. Asian Research Journal of Mathematics Vol 14 (3), (2019), 1-9.
• [5] Yuksel Soykan, Studies On the Recurrence Properties of Generalized Pentanacci Sequence, Journal of Progressive Research in Mathematics Vol 18(1), (2021) 64-71.
• [6] Yuksel Soykan, Nejla Ozmen and Melih Go¨o¨cen, On generalized Pentanacci quaternions, Tbilisi Math. J. Vol 13(4), (2020) 169-181.
• [7] B. Sivakumar and V. James, A Notes on Matrix Sequence of Pentanacci Numbers and Pentanacci Cubes,Vol. 13(2),(2022),603–611.DOI: 10.32513/tbilisi/ 1608606056
• [8] N. Sloane, The Encyclopedia of Integer Sequences, Elsevier Science Publishing Co., Inc.(1995).