TY  - JOUR
T1  - Pell Leonardo numbers and their matrix representations
AU  - Çelemoğlu, Çağla
PY  - 2024
DA  - August
Y2  - 2024
DO  - 10.54187/jnrs.1506171
JF  - Journal of New Results in Science
JO  - JNRS
PB  - Tokat Gaziosmanpasa University
WT  - DergiPark
SN  - 1304-7981
SP  - 101
EP  - 108
VL  - 13
IS  - 2
LA  - en
AB  - In this study, we investigate Pell numbers and Leonardo numbers and describe a new third-order number sequence entitled Pell Leonardo numbers. We then construct some identities, including the Binet formula, generating function, exponential generating function, Catalan, Cassini, and d’Ocagne’s identities for Pell Leonardo numbers and obtain a relation between Pell Leonardo and Pell numbers. In addition, we present some summation formulas of Pell Leonardo numbers based on Pell numbers. Finally, we create a matrix formula for Pell Leonardo numbers and obtain the determinant of the matrix.
KW  - Leonardo numbers
KW  - Pell numbers
KW  - Binet’s formula
KW  - Generating function
KW  - Matrix representation
CR  - M. Bicknell, A primer of the Pell sequence and related sequences, The Fibonacci Quarterly 13 (4) (1975) 345–349.
CR  - A. F. Horadam, Applications of modified Pell numbers to representations, Ulam Quarterly 3 (1) (1994) 34-53.
CR  - R. Melham, Sums involving Fibonacci and Pell numbers, Portugaliae Mathematica 56 (3) (1999) 309-318.
CR  - S. F. Santana, J. L. Díaz-Barrero, Some properties of sums involving Pell numbers, Missouri Journal of Mathematical Sciences 18 (1) (2006) 33-40.
CR  - Q. Mushtaq, U. Hayat, Pell numbers, Pell–Lucas numbers and modular group, In Algebra Colloquium 14 (1) (2007) 97-102.
CR  - A. Dasdemir, On the Pell, Pell-Lucas and modified Pell numbers by matrix method, Applied Mathematical Sciences 5 (64) (2011) 3173-3181.
CR  - S. Çelik, İ. Durukan, E. Özkan, New recurrences on Pell numbers, Pell-Lucas numbers, Jacobsthal numbers and Jacobsthal-Lucas numbers, Chaos, Solitons & Fractals 150 (2021), Article Number 111173 8 pages.
CR  - P. M. Catarino and A. Borges, On Leonardo numbers, Acta Mathematica Universitatis Comenianae 89 (1) (2019) 75-86.
CR  - A. G. Shannon, A note on generalized Leonardo numbers, Notes on Number Theory and Discrete Mathematics 25 (3) (2019) 97-101.
CR  - Y. Alp, E. G. Koçer, Some properties of Leonardo numbers, Konuralp Journal of Mathematics 9 (1) (2021) 183-189.
CR  - Y. Alp, E. G. Kocer, Hybrid Leonardo numbers, Chaos, Solitons & Fractals 150 (2021) Article Number 111128 5 pages.
CR  - K. Kuhapatanakul, J. Chobsorn, On the generalized Leonardo Numbers, Integers: Electronic Journal of Combinatorial Number Theory 22 (2022) Article Number A48 7 pages.
CR  - A. Karatas, On complex Leonardo numbers, Notes on Number Theory and Discrete Mathematics 28 (3) (2022) 458-465.
CR  - E. Tan, H. H. Leung, On Leonardo p-numbers, Integers: Electronic Journal of Combinatorial Number Theory 23 (2023) 1-11.
CR  - Y. Soykan, Generalized Horadam-Leonardo numbers and polynomials, Asian Journal of Advanced Research and Reports 17 (8) (2023) 128-169.
UR  - https://doi.org/10.54187/jnrs.1506171
L1  - https://dergipark.org.tr/en/download/article-file/4027421
ER  -