Repdigits in base b as product of two k-generalized Pell numbers

The k-Fibonacci sequence starts with the values (a total of k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we find all integers c having at least two representations as a difference between a k-Fibonacci number and a Pell number. This paper continues and extends the previous work of [J.J. Bravo, C.A. Gómez, and J.L. Herrera, On the intersection of k-Fibonacci and Pell numbers, Bull. Korean Math. Soc. 56(2) (2019), pp. 535–547; S. Hernández, F. Luca, and L.M. Rivera, On Pillai's problem with the Fibonacci and Pell sequences, Soc. Mat. Mex. 25 (2019), pp. 495–507 and M.O. Hernane, F. Luca, S.E. Rihane, and A. Togbé, On Pillai's problem with Pell numbers and powers of 2, Hardy- Ramanujan J. 41 (2018), pp. 22–31].

On the computing of the generalized order- k Pell numbers in log time

In this paper, we find all the Fibonacci numbers which are products of three Pell numbers and all Pell numbers which are products of three Fibonacci numbers.

In 2009, Luca and Nicolae [[Formula: see text], Integers 9 (2009) A30] proved that the only Fibonacci numbers whose Euler totient function is another Fibonacci number are [Formula: see text], and [Formula: see text]. In 2015, Faye and Luca [Pell numbers whose Euler function is a Pell number, Publ. Inst. Math. 101(115) (2017) 231–245] proved that the only Pell numbers whose Euler totient function is another Pell number are [Formula: see text] and [Formula: see text]. Here, we add to these two results and prove that for any fixed natural number [Formula: see text], if we define the sequence [Formula: see text] as [Formula: see text], [Formula: see text], and [Formula: see text] for all [Formula: see text], then the only solution to the Diophantine equation [Formula: see text] is [Formula: see text].

In this study, we investigate the form of the solutions of the following rational difference equation system x_n=(z_(n-1) z_(n-3))/(x_(n-2)+2z_(n-3) ),y_n=(x_(n-1) x_(n-3))/(〖-y〗_(n-2)+6x_(n-3) ),z_n=(y_(n-1) y_(n-3))/(z_(n-2)+14y_(n-3) ) ,n∈N_0 where initial values〖 x〗_(-3) 〖,x〗_(-2), x_(-1),y_(-3),y_(-2),y_(-1),〖 z〗_(-3),〖 z〗_(-2),〖 z〗_(-1) are nonzero real numbers, such that their solutions are associated with Pell numbers. We also give a relationships between Pell numbers and solutions of systems

We show that the only Pell numbers whose Euler function is also a Pell number are 1 and 2.

Falcón Santana and Díaz-Barrero [Missouri Journal of Mathematical Sciences, 18.1, pp. 33-40, 2006] proved that the sum of the first $4n+1$ Pell numbers is a perfect square for all $n \ge 0$. They also established two divisibility properties for sums of Pell numbers with odd index. In this paper, the sum of the first $n$ Pell numbers is characterized in terms of squares of Pell numbers for any $n \ge 0$. Additional divisibility properties for sums of Pell numbers with odd index are also presented, and divisibility properties for sums of Pell numbers with even index are derived.

In this study, we define a new generalization of the hybrid $k$-Pell sequence consisting of hybrid numbers with generalized hybrid $k$-Pell numbers as coefficients. We establish some algebraic properties and also the Binet formula, generating function, and exponential generating function related to this new sequence. In addition, some identities are provided as sum identities, and Catalan, Cassini, and d'Ocagne's identities. The particular cases are studied, namely, the hybrid numbers with hybrid $k$-Pell numbers as coefficients, the hybrid numbers with hybrid $k$-Pell--Lucas numbers as coefficients, and the hybrid numbers with hybrid Modified $k$-Pell numbers as coefficients.

In a recent note, Santana and Diaz-Barrero proved a number of sum identities involving the well-known Pell numbers. Their proofs relied heavily on the Binet formula for the Pell numbers. Our goal in this note is to reconsider these identities from a purely combinatorial viewpoint. We provide bijective proofs for each of the results by interpreting the Pell numbers as enumerators of certain types of tilings. In turn, our proofs provide helpful insight for straightforward generalizations of a number of the identities.

We obtain the Binet’s formula for k-Pell numbers and as a consequence we get some properties for k-Pell numbers. Also we give the generating function for k-Pell sequences and another expression for the general term of the sequence, using the ordinary generating function, is provided. Mathematics Subject Classification: 11B37, 05A15, 11B83.

Fibonacci, Lucas, and the ubiquitous Catalan numbers are delightful sources for experimentation, exploration, and conjecture (Askey 2005; Koshy 2001, 2006, 2008). Pell numbers are another similarly rich source. In this article, I will demonstrate some patterns associated with Pell numbers and then will show how they can be extracted from Pascal and Pascal–like triangles. I will provide a geometric interpretation of Pell numbers and conclude by citing a few opportunities for further exploration.

The infinite sum of reciprocal Pell numbers

Gaussian numbers means representation as Complex numbers. In this work, Gaussian Pell numbers are defined from recurrence relation of Pell numbers. Here the recurrence relation on Gaussian Pell number is represented in two dimensional approach. This provides an extension of Pell numbers into the complex plane.

The analytical study of the Pell number and the role of floor and ceiling functions into their computation is examined. Closed expressions of Pell numbers were initially derived using Binet's formula, followed by an asymptotic behavior study of the sequence using this formula. Taking into account the decreasing trend in the term $|\beta|^n = |1 - \sqrt{2}|^n$ for large values of $n$, a formula that closely approximates Pell numbers has been developed. In this context, relationships between numbers are clarified using floor and ceiling functions. The accuracy with which various theorems and lemmas mathematically prove these approximations is also included. The study also looks at limit processes with emphasis placed upon the determining influence that the ratio $\alpha = 1 + \sqrt{2}$ has on the growth rate of the sequence.

In this note we prove that for all positive integers $n$, the sum $S_{4n+1}$ of the first $4n+1$ Pell numbers is a perfect square. As a consequence, an identity involving binomial coefficients and Pell numbers is given. Also, sums of an even and odd number of terms of odd order are evaluated and some divisibility properties are obtained.