# New Types of Distance Padovan Sequences via Decomposition Technique
## 1 Introduction and Preliminary Results
It is worth mentioning that terms of Padovan sequence called Padovan numbers, usually denoted by \(\mathit{Pr}(n)\), were discovered in 1924 by a French student of architecture, [PERSON], and independently by a Dutch Benedictine monk and architect, [PERSON]. The name Padovan numbers was used by [PERSON] in honor of the contemporary architect [PERSON], who studied and popularized the plastic number well known as the following limit:  
\[\lim_{n\to\infty}\frac{\mathit{Pr}(n+1)}{\mathit{Pr}(n)}=1,3247\ldots. \tag{2}\]  
For historical details, see, for example, [17]. In the literature, we can find many generalizations of Padovan numbers in different contexts. We can meet them, for example, in the graph theory, as a result of counting sets of different types [18; 19], in combinatorics [20], in the theory of polynomials [21], and in cryptography, where properties of such sequences and new matrix generations used in encoding and decoding algorithms are particularly desired; see, for example, [22; 23]. In this paper, we consider three types of generalized distance Padovan numbers based on the same recurrence equation with distinct initial conditions that follow from special number decompositions. We derive direct formulas, prove some properties of the considered numbers, reveal their connection with [PERSON]'s triangle, and define new matrix generators for them. In [16], [PERSON] et al. introduced and studied distance Padovan numbers \(F_{3}(k,n)\) defined for integers \(k\geq 1\) and \(n\geq 0\) in the following way:  
\[F_{3}(k,n)=F_{3}(k,n-3)+F_{3}(k,n-k),\text{ for }n\geq max\{3,k\} \tag{3}\]  
with \(F_{3}(k,i)=1\), for \(i\in\{0,\ldots,max\{3,k\}-1\}\). Based on this idea we introduce other kinds of distance Padovan numbers as follows. Let \(k\geq 1\), \(n\geq 0\) be integers. Then, for a fixed \(j\in\{1,2,3\}\), distance Padovan numbers \(F_{3}^{(j)}(k,n)\) of the \(j\)-th kind are given by the recursion  
\[F_{3}^{(j)}(k,n)=F_{3}^{(j)}(k,n-3)+F_{3}^{(j)}(k,n-k)\text{ for }n\geq max\{3,k\} \tag{4}\]  
Now we give generating functions \(\sum_{n=0}^{\infty}F_{3}^{(j)}(k,n)x^{n}\) of the sequence \(\{F_{3}^{(j)}(k,n)\}\). Note that the generating function for \(\{F_{3}^{(2)}(k,n)\}\) was given by [PERSON] et al. in [16]. Theorem 1: [16] Let \(k\geq 1\), \(n\geq 0\) be integers. The generating function of \(\{F_{3}^{(2)}(k,n)\}\) has the following form:  
\[g^{(2)}(k,x)=\frac{1+t}{1-x^{3}-x^{k}},\text{ where }t= \begin{cases}0&\text{ for }k=1,\\ x&\text{ for }k=2,\\ x+x^{2}&\text{ for }k\geq 3.\end{cases}  \tag{8}\]  
Theorem 2: Let \(k\geq 1\), \(n\geq 0\) be integers. The generating function \(g^{(j)}(k,x)\) of the sequence \(\{F_{3}^{(j)}(k,n)\}\), for \(j\in\{1,3\}\) has the following form:  
\[g^{(1)}(k,x)=\frac{1}{1-x^{3}-x^{k}}, \tag{9}\]  
\[g^{(3)}(k,x)=\frac{x+x^{2}}{1-x^{3}-x^{k}}. \tag{10}\]  
Proof: Let \(g^{(j)}(k,x)=\sum\limits_{n=0}^{\infty}F_{3}^{(j)}(k,n)x^{n}\). Using the definition (4), we have  
\[g^{(j)}(k,x)-x^{3}g^{(j)}(k,x)-x^{k}g^{(j)}(k,x)=t,\text{ where }t= \begin{cases}1&\text{ for }F_{3}^{(1)}(k,n),\\ x+x^{2}&\text{ for }F_{3}^{(3)}(k,n),k\geq 2.\end{cases} \]  
Hence \(g^{(j)}(k,x)=\frac{t}{1-x^{3}-x^{k}}\), for \(j\in\{1,3\}\), which ends the proof.