Some Properties of Generalized Jacobsthal-Like Sequences

Yıl 2024, Cilt: 14 Sayı: 2, 92 - 96, 23.07.2024

Can Murat Dikmen , Kübra Karataş Selam

Öz

In this article, using Jacobsthal and Jacobsthal-Lucas sequences, we define generalized Jacobsthal-Like sequences and investigate their algebraic properties like Binet’s formula, generating functions, Simson formula and summation formula. We also prove some other summation formulas like sum of even and odd indices and alternating sum of generalized Jacobsthal-Like sequences.

Anahtar Kelimeler

Generalized Jacobsthal-Like sequences, Jacobsthal sequence, Jacobsthal-Lucas sequences

Proje Numarası

2

Genelleştirilmiş Jacobsthal-Benzeri Dizilerin Bazı Özellikleri

Öz

Bu makalede Jacobsthal ve Jacobsthal-Lucas dizilerini kullanarak genelleştirilmiş Jacobsthal-Benzeri dizilerini tanımlayıp Binet formülü, üreten fonksiyonlar, Simson formülü ve toplam formülü gibi cebirsel özelliklerini araştırıyoruz. Ayrıca çift ve tek indekslerin toplamı ve genelleştirilmiş Jacobsthal-Benzeri dizilerinin alterne toplamı gibi diğer toplama formüllerini de kanıtlıyoruz.

Anahtar Kelimeler

Genelleştirilmiş Jacobsthal-benzeri diziler, Jacobsthal dizisi, Jacobsthal-Lucas dizileri

Proje Numarası

2

Kaynakça

• Benjamin, AT., Quinn, JJ. 1999. Recounting Fibonacci and Lucas identities. The College Mathematics Journal, 30(5), 359-366. https://doi.org/10.1080/07468342.1999.11974086
• Badshah, VH., Teeth, MS., Dar, MM. 2012. Generalized Fibonacci-like sequence and its properties. International Journal of Contemporary Mathematical Sciences, 7(24), 1155-1164.
• Gupta, Y., Singh, M., Sikhwal, O. 2014. Generalized Fibonacci-like sequence associated with Fibonacci and Lucas sequences. Turkish Journal of Analysis and Number Theory, 2(6), 233-238. https://doi.org/10.12691/tjant-2-6-9
• Horadam, AF. 1996. Jacobsthal representation numbers. Fibonacci Quarterly, 34(1), 40-54.
• Harne, S., Singh, Pal, S. 2014. Generalized Fibonacci-Like sequence and Fibonacci Sequence. International Journal of Contemporary Mathematical Sciences, 9(5), 235-241. http://dx.doi.org/10.12988/ijcms.2014.4218
• Lee, JZ., Lee, JS. 1987. Some Properties of Generalization of the Fibonacci Sequences. The Fibonacci Quarterly, (No. 2), 110-117.
• Natividad, LR. 2016. Notes on Jacobsthal and Jacobsthal-Like Sequences. International Journal of Mathematics Trends and Technology (IJMTT), 34(2), 115-117. https://doi.org/10.14445/22315373/IJMTT-V34P519
• Pakapongpun, A. 2020. Identities on the product of Jacobsthal-Like and Jacobsthal-Lucas numbers. Notes on Number Theory and Discrete Mathematics, 26(1), 209-215. DOI: 10.7546/nntdm.2020.26.1.209-215
• Singh, B., Sikhwal, O., Bhatnagar, S. 2010. Fibonacci-Like Sequence and its properties. International Journal of Contemporary Mathematical Sciences, 5(18), 857-868. DOI: 10.12691/tjant-2-4-1.
• Singh, M., Sikhwal, O., Gupta, Y. 2014. Identities of generalized Fibonacci-Like Sequence. Turkish Journal of Analysis and Number Theory, 2(5), 170-175. DOI: 10.12691/tjant-2-5-3.
• Soykan, Y., Göcen, M. (2022). Binomial transform of the generalized third order Jacobsthal sequence. Asian-European Journal of Mathematics, 15(12). https://doi.org/10.1142/S1793557122502242
• Soykan, Y., Taşdemir, E., Okumuş, İ., Göcen, M. 2018. Gaussian generalized Tribonacci numbers. Journal of Progressive Research in Mathematics, 14(2), 2373-2387.