Öz
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.
Anahtar Kelimeler
Shifted Fibonacci Numbers Binet Formula Identities Summation Formulas
Kaynakça
- 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.
Öz
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.
Anahtar Kelimeler
Kaymış Fibonacci Sayıları Binet Formülü Özdeşlikler Toplam Formülleri
Kaynakça
- 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.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Cebir ve Sayı Teorisi |
| Bölüm | Makaleler |
| Yazarlar | |
| Erken Görünüm Tarihi | 22 Aralık 2023 |
| Yayımlanma Tarihi | 28 Aralık 2023 |
| Gönderilme Tarihi | 21 Ağustos 2023 |
| Yayımlandığı Sayı | Yıl 2023 Cilt: 23 Sayı: 6 |
