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    Binomial Transforms of k-Narayana Sequences and Some Properties
    (Fuat USTA, 2024) Kaplan, Faruk; Öztürk, Arzu Özkoç
    The aim of the study is to obtain new binomial transforms for the $k-$ Narayana sequence. The first of these is the binomial transform, which is its normal form, and in the first step, after finding the recurrence relation of this new binomial transform, the generating function and Binet formula were obtained. Finally, Pascal's triangle was calculated. In the rest of the article, $k-$binomial transform was performed for the $k-$ Narayana sequence and the recurrence relation, generating function, Binet formula and Pascal's triangle were examined for the new sequence obtained. Then, by performing the falling binomial transform and the rising binomial transform, the features listed above were found again for these sequences.
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    Cobalancing Numbers: Another Way of Demonstrating Their Properties
    (Emrah Evren KARA, 2024) Öztürk, Arzu Özkoç; Külahlı, Volkan
    In this study, previously obtained cobalancing numbers are considered from a different perspective, and the properties of the numbers are re-examined. The main purpose is to change the recurrence relation of cobalancing numbers and calculate some relations and properties in a more diverse and easier way. The reason that led us to this method is that the recurrence relation of cobalancing numbers has a second-order but non-homogeneous difference equations. Thus, it will be much easier to find the Binet formula, generating function, sum formulas, and many other relations with a sequence that is homogeneous and has a third-degree recurrence relation. Also some identities that have not been found before in the sequence are also included in this study.
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    On Gaussian Quadra Fibona-Pell Sequence and A Quaternion Sequence Formed by the Terms of This Sequence
    (Duzce University, 2025) Öztürk, Arzu Özkoç; Kaplan, Faruk
    In this study, the Gaussian quadra Fibona-Pell sequence is proposed and examined. The quadra Fibona-Pell sequence is first extended to define the Gaussian quadra Fibona-Pell sequence. Then the generating function, Binet-like formula, and some identities are represented. In addition, some formulas related to the Gaussian quadra Fibona-Pell sequence and some matrices containing terms of the sequence are studied. Finally we define a quaternion sequence formed by the terms of Gaussian quadra Fibona-Pell sequence.
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    On Generalized Tribonacci Octonions
    (2019) Öztürk, Arzu Özkoç
    In this paper, we introduce generalized tribonacci octonion sequence which is a generalizationof the third order recurrence relations. We investigate many identities which are created byusing generalized tribonacci sequence. We get different results for these classes of octonions,comprised recurrence relation, summation formulas, Binet formula, norm value and generatingfunction.