Abstract
Shifted Fibonacci numbers have been examined in the literature in terms of the greatest common divisor, but appropriate definitions and fundamental equations have not been worked on. In this article, we have obtained the Binet formula, which is a fundamental equation used to obtain the necessary element of the shifted Fibonacci number sequence. Additionally, we have obtained many well-known identities such as Cassini, Honsberger, and various other identities for this sequence. Furthermore, summation formulas for shifted Fibonacci numbers have been presented.
References
- Alp, Y., Kocer, E. G., 2021. Hybrid Leonardo numbers. Chaos, Solitons & Fractals, 150, 111128. Catarino, P. M., Borges, A., 2019. On Leonardo numbers. Acta Mathematica Universitatis Comenianae, 89, 75–86.
- Chen, K.-W., 2011. Greatest common divisors in shifted Fibonacci sequences. Journal of Integer Sequences, 14, 11.4.7.
- Dudley, U., Tucker, B., 1971. Greatest common divisors in altered Fibonacci sequences. Fibonacci Quarterly, 9, 89–91.
- Falcon, S., Plaza, A., 2007. On the Fibonacci k-numbers. Chaos, Solitons & Fractals, 32, 1615–1624. Halici, S., Karatas¸, A., 2017. Some matrix representations of Fibonacci quaternions and octonions. Advances in Applied Clifford Algebras, 27, 1233–1242.
- Hernandez, S., Luca, F., 2003. Common factors of shifted Fibonacci numbers. Periodica Mathematica Hungarica, 47, 95–110.
- Horadam, A., 1961. A generalized Fibonacci sequence. The American Mathematical Monthly, 68, 455–459.
- Horadam, A. F., 1963. Complex Fibonacci numbers and Fibonacci quaternions. The American Mathematical Monthly, 70, 289–291.
- Karatas¸, A., 2022. On complex Leonardo numbers. Notes on Number Theory and Discrete Mathematics, 28, 458–465.
- Nalli, A., Haukkanen, P., 2009. On generalized Fibonacci and Lucas polynomials. Chaos, Solitons & Fractals, 42, 3179–3186.
- Özkan, E., Aydoğdu, A., Altun, İ., 2017. Some identities for a family of Fibonacci and Lucas numbers. Journal of Mathematics and Statistical Science, 3, 295–303.
- Sanna, C., 2020. On the lcm of shifted Fibonacci numbers. arXiv preprint arXiv:(2007.13330). https://doi.org/10.48550/arXiv.2007.13330
- Sloane, N. J. A., 1973. A Handbook of Integer Sequences. Academic press, 40.
- Sloane, N. J. A., Plouffe, S. 1995. The encyclopedia of integer sequences. Academic press, 129.
- Spilker, J., Thang, L. B., Giay, C., 2022. The greatest common divisor of shifted Fibonacci numbers. Journal of Integer Sequences, 25, 22.1.7.
Abstract
Kaymış Fibonacci sayıları, literatürde, en büyük ortak bölen açısından incelenmiştir, ancak uygun tanım ve temel denklemler çalışılmamıştır. Bu makalede, kaymış Fibonacci sayı dizisinin gerekli elemanını elde etmek için kullanılan ve temel bir formül olan Binet formülünü verdik. Ayrıca, Cassini, Honsberger ve diğer birçok bilinen özdeşlikleri ve bu dizi için çok sayıda farklı özdeşlikler elde edilmiştir. Ayrıca, kaymış Fibonacci sayıları için toplama formülleri sunulmuştur.
References
- Alp, Y., Kocer, E. G., 2021. Hybrid Leonardo numbers. Chaos, Solitons & Fractals, 150, 111128. Catarino, P. M., Borges, A., 2019. On Leonardo numbers. Acta Mathematica Universitatis Comenianae, 89, 75–86.
- Chen, K.-W., 2011. Greatest common divisors in shifted Fibonacci sequences. Journal of Integer Sequences, 14, 11.4.7.
- Dudley, U., Tucker, B., 1971. Greatest common divisors in altered Fibonacci sequences. Fibonacci Quarterly, 9, 89–91.
- Falcon, S., Plaza, A., 2007. On the Fibonacci k-numbers. Chaos, Solitons & Fractals, 32, 1615–1624. Halici, S., Karatas¸, A., 2017. Some matrix representations of Fibonacci quaternions and octonions. Advances in Applied Clifford Algebras, 27, 1233–1242.
- Hernandez, S., Luca, F., 2003. Common factors of shifted Fibonacci numbers. Periodica Mathematica Hungarica, 47, 95–110.
- Horadam, A., 1961. A generalized Fibonacci sequence. The American Mathematical Monthly, 68, 455–459.
- Horadam, A. F., 1963. Complex Fibonacci numbers and Fibonacci quaternions. The American Mathematical Monthly, 70, 289–291.
- Karatas¸, A., 2022. On complex Leonardo numbers. Notes on Number Theory and Discrete Mathematics, 28, 458–465.
- Nalli, A., Haukkanen, P., 2009. On generalized Fibonacci and Lucas polynomials. Chaos, Solitons & Fractals, 42, 3179–3186.
- Özkan, E., Aydoğdu, A., Altun, İ., 2017. Some identities for a family of Fibonacci and Lucas numbers. Journal of Mathematics and Statistical Science, 3, 295–303.
- Sanna, C., 2020. On the lcm of shifted Fibonacci numbers. arXiv preprint arXiv:(2007.13330). https://doi.org/10.48550/arXiv.2007.13330
- Sloane, N. J. A., 1973. A Handbook of Integer Sequences. Academic press, 40.
- Sloane, N. J. A., Plouffe, S. 1995. The encyclopedia of integer sequences. Academic press, 129.
- Spilker, J., Thang, L. B., Giay, C., 2022. The greatest common divisor of shifted Fibonacci numbers. Journal of Integer Sequences, 25, 22.1.7.
Details
| Primary Language | English |
|---|---|
| Subjects | Algebra and Number Theory |
| Journal Section | Articles |
| Authors | |
| Early Pub Date | December 22, 2023 |
| Publication Date | December 28, 2023 |
| Submission Date | August 21, 2023 |
| Published in Issue | Year 2023 Volume: 23 Issue: 6 |
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