• Kyungpook Mathematical Journal
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• v.49 no.3
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• pp.483-492
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• 2009

In this paper we give formulae for sums of products of two Horadam type generalized Fibonacci numbers with the same recurrence equation and with possibly different initial conditions. Analogous improved alternating sums are also studied as well as various derived sums when terms are multiplied either by binomial coefficients or by members of the sequence of natural numbers. These formulae are related to the recent work of Belbachir and Bencherif, $\v{C}$erin and $\v{C}$erin and Gianella.

• Communications of the Korean Mathematical Society
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• v.33 no.3
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• pp.695-704
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• 2018

In this paper, firstly we compute the spectral norm of g-circulant matrices $C_{n,g}=g-Circ(c_0,c_1,{\cdots},c{_{n-1}})$, where $c_i{\geq}0$ or $c_i{\leq}0$ (equivalently $c_i{\cdot}c_j{\geq}0$). After, we compute the spectral norms, determinants and inverses of the g-circulant matrices with the Fibonacci and Lucas numbers.

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.1033-1048
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• 2020

We synthesize the recent work done on conditionally defined Lucas and Fibonacci numbers, tying together various definitions and results generalizing the linear recurrence relation. Allowing for any initial conditions, we determine the generating function and a Binet-like formula for the general sequence, in both the positive and negative directions, as well as relations among various sequence pairs. We also determine conditions for periodicity of these sequences and graph some recurrent figures in Python.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.3
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• pp.577-590
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• 2013

The work is devoted to study Fibonacci and tribonacci numbers. We study the modular formulas and the periods of the sequences.

• Honam Mathematical Journal
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• v.40 no.1
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• pp.139-153
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• 2018

Let $P{\geq}3$ be an integer and let ($U_n$) denote generalized Fibonacci sequence defined by $U_0=0$, $U_1=1$ and $U_{n+1}=PU_n-U_{n-1}$ for $n{\geq}1$. In this study, when P is odd, we solve the equation $U_n=11x^2+1$. We show that only $U_1$ and $U_2$ may be of the form $11x^2+1$.

• Honam Mathematical Journal
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• v.45 no.3
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• pp.418-432
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• 2023

In this study, we consider the Hankel and Gram matrices which are defined by the elements of special number sequences. Firstly, the eigenvalues, determinant, and norms of the Hankel matrix defined by special number sequences are obtained. Afterwards, using the relationship between the Gram and Hankel matrices, the eigenvalues, determinants, and norms of the Gram matrices defined by number sequences are given.

• Honam Mathematical Journal
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• v.44 no.4
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• pp.473-484
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• 2022

In this study, some new relations between generalized Fibonacci numbers and matrices are given. The work is designed in three stages: Firstly, it is obtained a relation between generalized Fibonacci numbers and integer powers of the matrices X satisfying the relation X² = pX +qI, and also, many results are derived from obtained relation. Then, it is established more general relation between generalized Fibonacci numbers and the square matrices X satisfying the condition X² = V_nX - (-q)ⁿI. Finally, some applications and numerical examples related to the obtained results are given.

• The Korean Journal of Applied Statistics
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• v.22 no.5
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• pp.1103-1113
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• 2009

We know that the first digits sequence of fibonacci numbers obey Benford's law. For the sequence in which the first two numbers are the arbitrary integers and the recurrence relation $a_{n+2}=a_{n+1}+a_n$ is satisfied, we can find that the first digits sequence of this sequence obey Benford's law. Also, we can find the stucture of the first digits sequence of this sequence with the exploratory data analysis tools.

• Journal of Periodontal and Implant Science
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• v.42 no.6
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• pp.217-223
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• 2012

Purpose: Several parameters have been described for determining the success or failure of dental implants. The surface properties of transgingival implant components have had a great impact on the long-term success of dental implants. The purpose of this study was to compare the tendency of two periodontal pathogens to adhere to and colonize zirconia abutments and titanium alloys both in hard surfaces and soft tissues. Methods: Twelve patients participated in this study. Three months after implant placement, the abutments were connected. Five weeks following the abutment connections, the abutments were removed, probing depth measurements were recorded, and gingival biopsies were performed. The abutments and gingival biopsies taken from the buccal gingiva were analyzed using real-time polymerase chain reaction to compare the DNA copy numbers of Aggregatibacter actinomycetemcomitans, Porphyromonas gingivalis, and total bacteria. The surface free energy of the abutments was calculated using the sessile water drop method before replacement. Data analyses used the Mann Whitney U-test, and P-values below 0.05 find statistical significance. Results: The present study showed no statistically significant differences between the DNA copy numbers of A. actinomycetemcomitans, P. gingivalis, and total bacteria for both the titanium and zirconia abutments and the biopsies taken from their buccal gingiva. The differences between the free surface energy of the abutments had no influence on the microbiological findings. Conclusions: Zirconia surfaces have comparable properties to titanium alloy surfaces and may be suitable and safe materials for the long-term success of dental implants.