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GENERALIZED FIBONACCI AND LUCAS NUMBERS OF THE FORM wx<sup>2</sup> AND wx<sup>2</sup> ∓ 1
Keskin, Refik Korean Mathematical Society 2014 대한수학회보 Vol.51 No.4
Let $P{\geq}3$ be an integer and let ($U_n$) and ($V_n$) denote generalized Fibonacci and Lucas sequences defined by $U_0=0$, $U_1=1$; $V_0= 2$, $V_1=P$, and $U_{n+1}=PU_n-U_{n-1}$, $V_{n+1}=PV_n-V_{n-1}$ for $n{\geq}1$. In this study, when P is odd, we solve the equations $V_n=kx^2$ and $V_n=2kx^2$ with k | P and k > 1. Then, when k | P and k > 1, we solve some other equations such as $U_n=kx^2$, $U_n=2kx^2$, $U_n=3kx^2$, $V_n=kx^2{\mp}1$, $V_n=2kx^2{\mp}1$, and $U_n=kx^2{\mp}1$. Moreover, when P is odd, we solve the equations $V_n=wx^2+1$ and $V_n=wx^2-1$ for w = 2, 3, 6. After that, we solve some Diophantine equations.
GENERALIZED FIBONACCI AND LUCAS NUMBERS OF THE FORM wx 2 AND wx 2 ∓ 1
Refik Keskin 대한수학회 2014 대한수학회보 Vol.51 No.4
Let P ≥ 3 be an integer and let (Un) and (Vn) denote gen- eralized Fibonacci and Lucas sequences defined by U0 = 0, U1 = 1; V0 = 2, V1 = P, and Un+1 = PUn − Un−1, Vn+1 = PVn − Vn−1 for n ≥ 1. In this study, when P is odd, we solve the equations Vn = kx2 and Vn = 2kx2 with k | P and k > 1. Then, when k | P and k > 1, we solve some other equations such as Un = kx2,Un = 2kx2,Un = 3kx2, Vn = kx2 ∓ 1, Vn = 2kx2 ∓1, and Un = kx2 ∓1. Moreover, when P is odd, we solve the equations Vn = wx2 +1 and Vn = wx2 −1 for w = 2, 3, 6. After that, we solve some Diophantine equations.
SOME NEW IDENTITIES CONCERNING THE HORADAM SEQUENCE AND ITS COMPANION SEQUENCE
Keskin, Refik,Siar, Zafer Korean Mathematical Society 2019 대한수학회논문집 Vol.34 No.1
Let a, b, P, and Q be real numbers with $PQ{\neq}0$ and $(a,b){\neq}(0,0)$. The Horadam sequence $\{W_n\}$ is defined by $W_0=a$, $W_1=b$ and $W_n=PW_{n-1}+QW_{n-2}$ for $n{\geq}2$. Let the sequence $\{X_n\}$ be defined by $X_n=W_{n+1}+QW_{n-1}$. In this study, we obtain some new identities between the Horadam sequence $\{W_n\}$ and the sequence $\{X_n\}$. By the help of these identities, we show that Diophantine equations such as $$x^2-Pxy-y^2={\pm}(b^2-Pab-a^2)(P^2+4),\\x^2-Pxy+y^2=-(b^2-Pab+a^2)(P^2-4),\\x^2-(P^2+4)y^2={\pm}4(b^2-Pab-a^2),$$ and $$x^2-(P^2-4)y^2=4(b^2-Pab+a^2)$$ have infinitely many integer solutions x and y, where a, b, and P are integers. Lastly, we make an application of the sequences $\{W_n\}$ and $\{X_n\}$ to trigonometric functions and get some new angle addition formulas such as $${\sin}\;r{\theta}\;{\sin}(m+n+r){\theta}={\sin}(m+r){\theta}\;{\sin}(n+r){\theta}-{\sin}\;m{\theta}\;{\sin}\;n{\theta},\\{\cos}\;r{\theta}\;{\cos}(m+n+r){\theta}={\cos}(m+r){\theta}\;{\cos}(n+r){\theta}-{\sin}\;m{\theta}\;{\sin}\;n{\theta},$$ and $${\cos}\;r{\theta}\;{\sin}(m+n){\theta}={\cos}(n+r){\theta}\;{\sin}\;m{\theta}+{\cos}(m-r){\theta}\;{\sin}\;n{\theta}$$.
GENERALIZED LUCAS NUMBERS OF THE FORM 5kx2 AND 7kx2
Olcay Karaatli,Refik Keskin 대한수학회 2015 대한수학회보 Vol.52 No.5
Generalized Fibonacci and Lucas sequences (Un) and (Vn) are defined by the recurrence relations Un+1 = PUn+QUn−1 and Vn+1 = PVn +QVn−1, n ≥ 1, with initial conditions U0 = 0, U1 = 1 and V0 = 2, V1 = P. This paper deals with Fibonacci and Lucas numbers of the form Un(P,Q) and Vn(P,Q) with the special consideration that P ≥ 3 is odd and Q = −1. Under these consideration, we solve the equations Vn = 5kx2, Vn = 7kx2, Vn = 5kx2±1, and Vn = 7kx2±1 when k | P with k > 1. Moreover, we solve the equations Vn = 5x2 ±1 and Vn = 7x2 ± 1.
GENERALIZED FIBONACCI NUMBERS OF THE FORM 11x<sup>2</sup> + 1
Ogut, Ummugulsum,Keskin, Refik The Honam Mathematical Society 2018 호남수학학술지 Vol.40 No.1
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 $11x^2+1$.
GENERALIZED FIBONACCI NUMBERS OF THE FORM 11x^2 + 1
Ummugulsum Ogut,Refik Keskin 호남수학회 2018 호남수학학술지 Vol.40 No.1
Let P 3 be an integer and let (Un) denote generalizedFibonacci sequence de ned by U0 = 0;U1 = 1 and Un+1 = PUn Un 1 for n 1: In this study, when P is odd, we solve the equationUn = 11x2 + 1. We show that only U1 and U2 may be of the form11x2 + 1:
Repdigits as difference of two Pell or Pell-Lucas numbers
Fatih Erduvan,Refik Keskin 강원경기수학회 2023 한국수학논문집 Vol.31 No.1
In this paper, we determine all repdigits, which are difference of two Pell and Pell-Lucas numbers. It is shown that the largest repdigit which is difference of two Pell numbers is $99=169-70=P_{7}-P_{6}$ and the largest repdigit which is difference of two Pell-Lucas numbers is $444=478-34=Q_{7}-Q_{4}.$
GENERALIZED LUCAS NUMBERS OF THE FORM 5kx<sup>2</sup> AND 7kx<sup>2</sup>
KARAATLI, OLCAY,KESKIN, REFIK Korean Mathematical Society 2015 대한수학회보 Vol.52 No.5
Generalized Fibonacci and Lucas sequences ($U_n$) and ($V_n$) are defined by the recurrence relations $U_{n+1}=PU_n+QU_{n-1}$ and $V_{n+1}=PV_n+QV_{n-1}$, $n{\geq}1$, with initial conditions $U_0=0$, $U_1=1$ and $V_0=2$, $V_1=P$. This paper deals with Fibonacci and Lucas numbers of the form $U_n$(P, Q) and $V_n$(P, Q) with the special consideration that $P{\geq}3$ is odd and Q = -1. Under these consideration, we solve the equations $V_n=5kx^2$, $V_n=7kx^2$, $V_n=5kx^2{\pm}1$, and $V_n=7kx^2{\pm}1$ when $k{\mid}P$ with k > 1. Moreover, we solve the equations $V_n=5x^2{\pm}1$ and $V_n=7x^2{\pm}1$.
GENERALIZED FIBONACCI NUMBERS OF THE FORM 11x<sup>2</sup> + 1
( Ummugulsum Ogut ),( Refik Keskin ) 호남수학회 2018 호남수학학술지 Vol.40 No.1
Let P ≥ 3 be an integer and let (Un) denote generalized Fibonacci sequence defined by U0 = 0,U<sub>1</sub> = 1 and U<sub>n+1</sub> = PUn - U<sub>n-1</sub> for n ≥ 1, In this study, when P is odd, we solve the equation Un = 11x<sup>2</sup> + 1. We show that only U<sub>1</sub> and U<sub>2</sub> may be of the form 11x<sup>2</sup> + 1.
SUM FORMULAE OF GENERALIZED FIBONACCI AND LUCAS NUMBERS
Cerin, Zvonko,Bitim, Bahar Demirturk,Keskin, Refik The Honam Mathematical Society 2018 호남수학학술지 Vol.40 No.1
In this paper we obtain some formulae for several sums of generalized Fibonacci numbers $U_n$ and generalized Lucas numbers $V_n$ and their dual forms $G_n$ and $H_n$ by using extensions of an interesting identity by A. R. Amini for Fibonacci numbers to these four kinds of generalizations and their first and second derivatives.