相关论文: Narayana's cows numbers which are concatenations o…

Let $ (N_n)_{n\ge 0}$ be the Narayana's cow sequence defined by a third-order recurrence relation $ N_0=0,\ N_1= N_2=1 $, and $ N_{n+3}=N_{n+2}+N_n $ for all $ n\ge 0 $. In this paper, we determine all Narayana numbers that are…

Narayana's cows sequence satisfies the third-order linear recurrence relation $N_n=N_{n-1}+N_{n-3}$ for $n \geq 3$ with initial conditions $N_0=0$ and $N_1=N_2=1$. In this paper, we study $b$-repdigits which are sums of two Narayana…

Let $(N_{n})_{n\ge 0}$ be Narayana's cows sequence given by a recurrence relation $ N_{n+3}=N_{n+2}+N_n $ for all $ n\ge 0 $, with initial conditions $ N_0=0 $, and $ N_1= N_2=1 $. In this paper, we find all members in Narayana's cow…

Narayana's cows problem is a problem similar to the Fibonacci's rabbit problem. We define the numbers which are the solutions of this problem as Narayana's cows numbers. Narayana's cows sequence satisfies the third order recurrence relation…

In this paper, we show that there are only finitely many Narayana's numbers which can be written as product of three repdigits in base $g$ with $g \geq 2$. Moreover, for $2 \leq g \leq 10$, we determine all these numbers.

We define the Narayana sequence $\{a_n\}_{n\geq 0}$ as the one satisfying the linear recurrence relation $a_n = a_{n-1}+a_{n-3}$ for $n\geq 3$, with initial values $a_0 = 0$ and $a_1 = a_2=1$. In this paper, we fully characterize the…

In this paper, we focus on Narayana numbers which can be written as a products of four repdigits in base $g$, where $g$ is an integer with $g\geq2$. We prove that for $g$ between $2$ and $12$, there are finitely many of these numbers.…

Let $\{N_m\}_{m\ge0}$ be the Narayana's cows sequence given by $N_0=0$, $N_1=1=N_2=1$ and \[ N_{m+3}=N_{m+2}+N_m,\quad \text{ for }\; m\geq 0 \] and let $\{F_n\}_{n\ge0}$ be the Fibonacci sequence. In this paper we solve explicitely the…

In this paper, we find all repdigits which can be expressed as the product of a Narayana, and a product of two repdigits is Narayana.

Let $\left\lbrace a_{n}\right\rbrace_{n\geq 0}$ be the Narayana Sequence defined by the recurence $a_{n}=a_{n-1}+a_{n-3}$ for all $n\geq 3$ with intital values $a_{0}=0$ and $a_{1}=a_{2}=1$. In This paper, we fully characterize the $3-$adic…

In the present paper we find a new interpretation of Narayana polynomials N_n(x) which are the generating polynomials for the Narayana numbers N_{n,k} counting Dyck paths of length n and with exactly k peaks. Strangely enough Narayana…

In this paper,we find all generalized Fibonacci numbers which are Narayana's cows numbers. In our proofs, we use both Baker's theory of nonzero linear forms in logarithms of algebraic numbers and the Baker-Davenport reduction method.

For the Fibonacci numbers $F_n$, we have the self-convolution formula $5 \sum_{i=0}^n F_i F_{n-i} = (2n)F_{n+1} - (n+1)F_n$. We find the corresponding self-convolution formula for the Narayana numbers $R_n$ which satisfy $R_n = R_{n-1} +…

The Lucas polynomials, $\{n\}$, are polynomials in $s$ and $t$ given by $\{ n \} = s \{ n-1 \} + t \{ n-2 \}$ for $n \geq 2$ with $ \{ 0 \} = 0$ and $\{ 1 \} = 1$. The lucanomial coefficients, an analogue of the binomial coefficients, are…

Padovan sequence is a ternary recurrent sequence defined by the recurrence relation $P_{n+3}=P_{n+1}+P_{n}$ with initial terms $P_{0}=P_{1}=P_{2}=1.$ In this study it is shown that $114,151,200,265,351,465,616,816,3329,4410,7739,922111$ are…

Let $ (P_n)_{n\ge 0}$ be the sequence of Perrin numbers defined by ternary relation $ P_0=3 $, $ P_1=0 $, $ P_2=2 $, and $ P_{n+3}=P_{n+1}+P_n $ for all $ n\ge 0 $. In this paper, we use Baker's theory for nonzero linear forms in logarithms…

Let $\mathcal{P}_{n}$ be the $n$-th Padovan number, $E_{n}$ be the $n$-th Perrin number and $N_{n}$ be the $n$-th Narayana's cows number. Let $b$ be a positive integer such that $b \geq 2$. In this paper, we study the Diophantine equations…

Let $\beta$ be a non-unit real algebraic integer greater than one and $\{a_{n}\}_{n \geq 0}$ be a sequence satisfying a linear recurrence relation $a_{n+3}=aa_{n+2}+ba_{n+1}+ca_{n}$. Under certain conditions, we prove that the number of…

Let $ \{P_{n}\}_{n\geq 0} $ be the sequence of Padovan numbers defined by $ P_0=0 $, $ P_1 =1=P_2$ and $ P_{n+3}= P_{n+1} +P_n$ for all $ n\geq 0 $. In this paper, we find all repdigits in base $ 10 $ which can be written as a sum of three…

A nondecreasing sequence of positive integers is $(\alpha,\beta)$-Conolly, or Conolly-like for short, if for every positive integer $m$ the number of times that $m$ occurs in the sequence is $\alpha + \beta r_m$, where $r_m$ is $1$ plus the…