# New Types of Distance Padovan Sequences via Decomposition Technique
## 4 Determinant Generators  
Matrix generators of Fibonacci-type sequences have been studied by many authors. In a number of papers we can find Padovan generators and related matrices in which terms of Padovan sequences are obtained by \(Q\)-matrices, determinants, or permanents; see, for example, results given by [PERSON] [25; 26]. In this paper, we give determinant generators for distance Padovan numbers \(F_{3}^{(j)}(k,n)\), \(j\in\{1,2,3\}\), constructed on a basis of a matrix of initial conditions. By a matrix of initial conditions of the sequence \(\{F_{3}^{(j)}(k,n)\}\) we mean a matrix \(A_{k}^{(j)}\) of size \(m\times m\), where \(m=max\{3,k\}\), such that consecutive main corner minors \(\det A_{k,i}^{(j)}\) of the matrix \(A_{k}^{(j)}\) are equal \(F_{3}^{(j)}(k,i-1)\), i.e.,  
\[A_{k}^{(j)}=[a_{ij}]_{m\times m}\text{ and }\det A_{k,i}^{(j)}= \begin{vmatrix}a_{ 11}&\ldots&a_{1i}\\ \vdots&&\vdots\\ a_{i1}&\ldots&a_{ii}\end{vmatrix} =F_{3}^{(j)}(k,i-1)\text{ for }i\in\{1,\ldots,m\}.\]  
Clearly, the matrix of initial conditions is not determined uniquely. At the beginning we discuss cases \(k\in\{1,2,3\}\). For \(k=1\) we set the following matrices of initial conditions \(A_{1}^{(1)}=A_{1}^{(2)}=A_{1}^{(3)}= \begin{bmatrix}1&0&1\\ 0&1&0\\ 0&1&1\end{bmatrix} \), for \(k=2\) we choose matrices  
\[A_{2}^{(1)}= \begin{bmatrix}1&1&1\\ 1&1&0\\ 0&1&1\end{bmatrix} ,\quad A_{2}^{(2)}= \begin{bmatrix}1&0&1\\ 1&1&1\\ 0&1&1\end{bmatrix} ,\quad A_{2}^{(3)}= \begin{bmatrix}0&-1&-1\\ 1&1&1\\ 0&1&1\end{bmatrix} ,\]  
and for \(k=3\) we set  
\[A_{3}^{(1)}= \begin{bmatrix}1&1&1\\ 1&1&1\\ 0&1&1\end{bmatrix} ,\quad A_{3}^{(2)}= \begin{bmatrix}1&0&1\\ 1&1&1\\ 0&1&1\end{bmatrix} ,\quad A_{3}^{(3)}= \begin{bmatrix}0&-1&0\\ 1&1&1\\ 0&1&1\end{bmatrix} .\]We construct a determinant generator \(A_{k,n}^{(j)}\) for \(F_{3}^{(j)}(k,n)\) with \(k\in\{1,2,3\}\) and \(j\in\{1,2,3\}\) starting from the matrix \(A_{k}^{(j)}\) by adding rows and columns according to the rule presented below. \[A_{1,n}^{(j)}=\left[ \begin{array}{ccccc}&0&\ldots\\ A_{1}^{(j)}&&1&\ddots\\ &&0&\ddots\\ \hline 0&0&1&1&\ddots\\ \vdots&\vdots&\ddots&\ddots&\ddots\\ \end{array} \right]_{n\times n,t}\quad A_{2,n}^{(j)}=\left[ \begin{array}{ cccccc}&0&\ldots\\ A_{2}^{(j)}&&1&\ddots\\ &&-1&\ddots\\ \hline 0&0&1&0&\ddots\\ \vdots&\vdots&\ddots&\ddots&\ddots\\ \end{array} \right]_{n\times n,t}\]  
**Theorem 9**. : _Let \(k\in\{1,2\}\), \(n\geq 1\) be integers. Then_  
_(i)_ \(F_{3}^{(j)}(1,n-1)=det\,A_{1,n}^{(j)}\) _for_ \(j\in\{1,2\}\)_,_  
_(ii)_ \(F_{3}^{(j)}(2,n-1)=det\,A_{2,n}^{(j)}\) _for_ \(j\in\{1,2,3\}\)_,_  
_(iii)_ \(F_{3}^{(j)}(3,n-1)=det\,A_{3,n}^{(j)}\) _for_ \(j\in\{1,2,3\}\)_._  
Proof. : (by induction on n)  
We will prove (i) for \(j=1\). The remaining cases can be proved analogously. If \(n\leq 3\), then by the definition of a matrix of initial conditions we have  
\[\det A_{1,n}^{(1)}= \begin{vmatrix}1&0&1&0\\ 1&1&1&1\\ 0&1&1&0\\ 0&0&1&1\end{vmatrix} = \begin{vmatrix}1&0&1\\ 1&1&1\\ 0&1&1\end{vmatrix} + \begin{vmatrix}1&0&1\\ 0&1&1\\ 0&0&1\end{vmatrix} = \begin{vmatrix}1&0&1\\ 1&1&1\\ 0&1&1\end{vmatrix} + \begin{vmatrix}1&0&1\\ 1&1&1\\ 0&1&1\end{vmatrix} + \begin{vmatrix}1&1\\ 1&1&1\\ 0&1&1\end{vmatrix} + \begin{vmatrix}1&1\\ 1&1\\ 0&1&1\end{vmatrix} + \begin{vmatrix}1\\ 1&1\\ 0&1&1\end{vmatrix} + \begin{vmatrix}1\\ 1&1\\ 0&1&1\end{vmatrix} + \begin{vmatrix}1\\ 1&1\\ 0&1&1\end{vmatrix} + \begin{vmatrix}1\\ 1&1\\ 0&1&1\end{vmatrix} + \begin{vmatrix}1\\ 1&1\\ 0&1&1\end{vmatrix} +\begin{vmatrix}1\\ 1&1  
\[A_{k}^{(1)}=\left[ \begin{array}{ccccc}&&&0&\ldots\\ &&&0&\ldots\\ &&&1&\ddots\\ &&&0&\ddots\\ \hline 0&0&0&1&0&\ddots\\ \vdots&\vdots&\ddots&\ddots&\ddots&\ddots\\ \end{array} \right]_{k\times k.} \tag{15}\]  
The coefficients of the matrix \(A_{k}^{(1)}=[a_{ij}]_{k\times k}\) can be described in the form  
\[a_{ij}= \begin{cases}1&\text{for }\ j=i\ \text{ where }i\in\{1,3\},\\ &j=i+2\ \text{ where }i\in\{2,\ldots,k-2\},\\ &j=i-1\ \text{ where }i\in\{3,\ldots,k\},\\ 0&\text{otherwise}.\end{cases}