Properties and Generalizations of Altered Jacobsthal Numbers Squared and their GCD Sequences
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Keywords:
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Altered Jacobsthal Number, Greatest Common Divisor (GCD) Sequence, Jacobsthal Sequence
Abstract
This paper investigates two types of altered Jacobsthal numbers, namely G(2)J(n) (a) and H(2)J(n) (a), which are obtained by adding or subtracting a specific value, denoted with {a}, from the square of the nth Jacobsthal numbers. These numbers exhibit a close relationship with the consecutive products of the Jacobsthal numbers. The study establishes consecutive sum-subtraction relations for the altered Jacobsthal numbers, and derives their Binet-like formulas. Furthermore, the greatest common divisor (Gcd) sequences of r-successive terms, represented by {G(2)J(n),r (a)} and {H(2)J(n),r (a)}, r ∈ {1, 2, 3, 4} are investigated. It is observed that these sequences display either a periodic or Jacobsthal structure.
References
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Bilgici, G. and D. Bród (2023). On r -Jacobsthal and r -Jacobsthal-Lucas Numbers. Annales Mathematicae Silesianae, 37(1); 16–31
Brod, D. (2020). On a new Jacobsthal-type sequence. Ars Combinatoria, 150; 21–29
Catarino, P. and A. Borges (2019). On Leonardo numbers. Acta Mathematica Universitatis Comenianae, 89(1); 75–86
Catarino, P., P. Vasco, H. Campos, A. P. Aires, and A. Borges (2015). New families of Jacobsthal and Jacobsthal-Lucas numbers. Algebra and Discrete Mathematics, 20(1)
Erduvan, F. and R. Keskin (2021). Fibonacci Numbers Which Are Products of Two Jacobsthal Numbers. Tbilisi Mathematical Journal, 14(2); 105–116
Flórez, R., R. A. Higuita, and L. Junes (2014a). 9-Modularity and GCD Properties of Generalized Fibonacci Numbers. Integers, 14; A55
Flórez, R., R. A. Higuita, and L. Junes (2014b). GCD Property of the Generalized Star of David in the Generalized Hosoya Triangle. Journal of Integer Sequences, 17(3); 14–3
Flórez, R., R. A. Higuita, and A. Mukherjee (2014c). Alternating Sums in the Hosoya Polynomial Triangle. Journal of Integer Sequences, 17(9); 14–9
Flórez, R. and L. Junes (2012). GCD Properties in Hosoya’s Triangle. The Fibonacci Quarterly, 50(2); 163–174
Flórez, R., R. A. Higuita, and A. Mukherjee (2018a). Characterization of the Strong Divisibility Property for Generalized Fibonacci Polynomials. Integers, 18; 14
Flórez, R., R. A. Higuita, and A. Mukherjee (2018b). The Star of David and Other Patterns in Hosoya Polynomial Triangles. Journal of Integer Sequences, 21(4); 18–4
Hillman, A. P. and V. E. Hoggatt (1972). A Proof of Gould’s Pascal Hexagon Conjecture. The Fibonacci Quarterly, 10(6); 565–598
Hoggatt, V. E. and W. Hansell (1971). The Hidden Hexagon Squares. The Fibonacci Quarterly, 9(2); 120–133
Horadam, A. F. (1988). Jacobsthal and Pell Curves. The Fibonacci Quarterly, 26(1); 77–83
Horadam, A. F. (1993). Associated Sequences of General Order. The Fibonacci Quarterly, 31(2); 165–165
Horadam, A. F. (1996a). Generalized Jacobsthal Representation Sequence {Tn}. Notes on Number Theory and Discrete Mathematics, 2(3); 5–11
Horadam, A. F. (1996b). Jacobsthal Representation Numbers. The Fibonacci Quarterly, 34(1); 40–54
Hosoya, H. (1976). The Fibonacci Quarterly. Fibonacci Triangle, 14; 173–178
Koken, F. (2019). Some Properties of The Altered Jacobsthal And Jacobsthal Lucas Numbers. International Marmara Science and Social Sciences Congress; 521–528
Komatsu, T. and C. Pita-Ruiz (2023). The Frobenius Number for Jacobsthal Triples Associated with Number of Solutions. Axioms, 12(2); 98
Koshy, T. (2019). Fibonacci and Lucas Numbers with Applications, volume 2. John Wiley & Sons
Sloane, N. J. A. (2003). The On-Line Encyclopedia of Integer Sequences (OEIS). Notices of American Mathematical Society, 50(8); 912–915
Uygun, S. (2021). The Relations between Bi-Periodic Jacobsthal and Bi-Periodic Jacobsthal Lucas Sequence. Cumhuriyet Science Journal, 42(2); 346–357
Uygun, S. and E. Owusu (2016). A New Generalization of Jacobsthal Numbers (bi-Periodic Jacobsthal Sequences). Journal of Mathematical Analysis, 7(5); 28–39
Wamiliana, W., S. Suharsono, and P. E. Kristanto (2019). Counting the Sum of Cubes for Lucas and Gibonacci Numbers. Science and Technology Indonesia, 4(2); 31–35
Wolfram, S. (1983). Statistical Mechanics of Cellular Automata. Reviews of Modern Physics, 55(3); 601–644
Wolfram, S. and M. Gad-el Hak (2003). A New Kind of Science. Applied Mechanics Reviews, 56(2); B18–B19
Yazlik, Y., N. Yilmaz, N. Taskara, and K. Uslu (2016). Jacobsthal Family Modulo M. TWMS Journal of Applied and Engineering Mathematics, 6(1); 15–21
Yuliana, A. (2023). The Relationship of Multiset, Stirling Number, Bell Number, and Catalan Number. Science and Technology Indonesia, 8(2); 330–337
Authors
Koken, F., Ergun, H., & Uzun, Y. (2025). Properties and Generalizations of Altered Jacobsthal Numbers Squared and their GCD Sequences. Science and Technology Indonesia, 10(1), 273–282. https://doi.org/10.26554/sti.2025.10.1.273-282
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