In an earlier paper it was argued that two sequences, denoted by {Un} and {Wn}, constitute the sextic analogues of the well-known Lucas sequences {un} and {vn}. While a number of the properties of {Un} and {Wn} were presented, several arithmetic properties of these sequences were only mentioned in passing. In this paper we discuss the derived sequences {Dn} and {En}, where Dn = gcd(Wn − 6R n, U...

The functional equation f(3x) = 4f(3x−3)+f(3x− 6) will be solved and its Hyers-Ulam stability will be also investigated in the class of functions f : R → X , where X is a real Banach space. Keywords—Functional equation, Lucas sequence of the first kind, Hyers-Ulam stability.

In this paper we find all solutions of four kinds of the Diophantine equations begin{equation*} ~x^{2}pm V_{t}xy-y^{2}pm x=0text{ and}~x^{2}pm V_{t}xy-y^{2}pm y=0, end{equation*}% for an odd number $t$, and, begin{equation*} ~x^{2}pm V_{t}xy+y^{2}-x=0text{ and}text{ }x^{2}pm V_{t}xy+y^{2}-y=0, end{equation*}% for an even number $t$, where $V_{n}$ is a generalized Lucas number. This pape...

Let $\lbrace U_n\rbrace =\lbrace U_n(P,Q)\rbrace $ and V_n\rbrace V_n(P,Q)\rbrace be the Lucas sequences of first second kind respectively at parameters $P \ge 1$ $Q \in \lbrace -1, 1\rbrace $. In this paper, we provide a technique for characterizing solutions so-called Bartz-Marlewski equation \[ x^2-3xy+y^2+x=0\,, \] where $(x,y)=(U_i, U_j)$ or $(V_i, V_j)$ with $i$, j 1$. Then, procedure is ...

Let L denote the n th Lucas number, where n is a natural number. n 2 Using elementary techniques, we find all solutions of the equation: Ln px where p is prime and p (i000. KEY Wi’RDS AND PHRASES. Lucas number 1985 AMS SUBJECT CLASSIFICATION CODE. IIB39 1. NTRODUCTION Let. n denote a natural number. Let L denote the n th Lucas number, that is, n L1=1, L2=3, Ln Ln_l+Ln_2 for n_ 3. In [1], J.H.E....

Porous media can be characterized by studying the kinetics of liquid rise within the pore spaces. Although porous media generally have a complex structure, they can be modeled as a single, vertical capillary or as an assembly of such capillaries. The main difficulties lie in separately estimating the effective mean radius of the capillaries and the contact angle between the liquid and the pore....

We derive an analytic solution for the capillary rise of liquids in a cylindrical tube or a porous medium in terms of height h as a function of time t. The implicit t(h) solution by Washburn is the basis for these calculations and the Lambert W function is used for its mathematical rearrangement. The original equation is derived out of the 1D momentum conservation equation and features viscous ...

In this paper, we give some determinantal and permanental representations of generalized Lucas polynomials, which are a general form of generalized bivariate Lucas p-polynomials, ordinary Lucas and Perrin sequences etc., by using various Hessenberg matrices. In addition, we show that determinant and permanent of these Hessenberg matrices can be obtained by using combinations. Then we show, the ...

Balancing numbers n and balancers r are solutions of the Diophantine equation 1 + 2 + . . . + (n 1) = (n + 1) + (n + 2) + . . . + (n + r). It is well-known that if n is a balancing number, then 8n2 + 1 is a perfect square and its positive square root is called a Lucas-balancing number. In this paper, some new identities involving balancing and Lucas-balancing numbers are obtained. Some divisibi...

an artificial neural network (ann) was used to analyse the capillary rise in porous media. wetting experiments were performed with fifteen liquids and fifteen different powders. the liquids covered a wide range  of  surface  tension ( 15.45-71.99  mj/m2 )  and  viscosity (0.25-21 mpa.s). the powders also provided an acceptable range of particle size (0.012-45 μm) and surface free energy (25.54-...