# New Types of Distance Padovan Sequences via Decomposition Technique
## 2 Number Decompositions  
Now we give an interpretation of distance Padovan numbers \(F_{3}^{(j)}(k,n)\) for \(j\in\{1,2,3\}\) and \(k\
eq 3\) with respect to a special kind of decompositions of a number \(n\). For combinatorial concepts not defined here, see [24]. Recall that for \(n\geq 1\) being an integer, a decomposition of the number \(n\) is its ordered partition. In other words, a decomposition of \(n\) is a t-tuple \((n_{1},\ldots,n_{t})\), where \(1\leq n_{i}\leq n\), \(i\in\{1,\ldots,t\}\), \(t\geq 1\), and \(\sum\limits_{i=1}^{t}n_{i}=n\). For example, \((4),(3,1),(1,3)(2,2),(1,1,2),(1,2,1),(2,1,1),(1,1,1,1)\) are different decompositions of the number \(4\). Let \(k\geq 1\), \(k\
eq 3\), and \(n\geq k-1\) be integers. A t-tuple \((n_{1},\ldots,n_{t})\) is called a \((3,k)\)-decomposition of the number \(n\) if \(n_{i}\in\{3,k\}\) for \(i\in\{1,\ldots,t\}\). We say that \((n_{1},\ldots,n_{t})\) is a \((3,k)_{r^{+}}\)-decomposition of the number \(n\) if \(n_{i}\in\{3,k\}\) for \(i\in\{1,\ldots,t-1\}\), and \(n_{t}=r\), where \(1\leq r\leq min\{3,k\}-1\). Similarly, we say that \((n_{1},\ldots,n_{t})\) is a \((3,k)_{r^{-}}\)-decomposition of the number \(n\) if \(n_{1}=r\), and \(n_{i}\in\{3,k\}\) for \(i\in\{2,\ldots,t\}\). Let us introduce the following decompositions' parameters:  
\(\gamma_{0}(3,k,n)\) - the number of all \((3,k)\)-decompositions;  
\(\gamma_{r^{-}}(3,k,n)\) - the number of all \((3,k)_{r^{-}}\)-decompositions;  
\(\gamma_{r^{+}}(3,k,n)\) - the number of all \((3,k)_{r^{+}}\)-decompositions;  
\(\gamma_{\leq r^{-}}(3,k,n)\) - the number of all \((3,k)\) and all \((3,k)_{s^{-}}\)-decompositions,  
where \(1\leq s\leq r\);  
\(\gamma_{\leq r^{+}}(3,k,n)\) - the number of all \((3,k)\) and all \((3,k)_{s^{+}}\)-decompositions,  
where \(1\leq s\leq r\);  
\(\gamma(3,k,n)\) - the number of all \((3,k)\), \((3,k)_{r^{-}}\) and \((3,k)_{r^{+}}\)-decomposition,  
where \(1\leq r\leq min\{3,k\}-1\). Clearly,  
\(\gamma_{r^{-}}(3,k,n)=\gamma_{r^{+}}(3,k,n)\),  
\(\gamma_{\leq r^{-}}(3,k,n)\)Moreover, if \(k>3,\ r=min\{3,k\}-1\), then  
\[\gamma_{\leq r^{-}}(3,k,n)=\gamma_{0}(3,k,n)+\sum\limits_{r=1}^{2} \gamma_{r^{-}}(3,k,n),\] \[\gamma_{\leq r^{+}}(3,k,n)=\gamma_{0}(3,k,n)+\sum\limits_{r=1}^{2} \gamma_{r^{+}}(3,k,n),\] \[\gamma(3,k,n)=\gamma_{0}(3,k,n)+\sum\limits_{r=1}^{2}(\gamma_{r^{ -}}(3,k,n)+\gamma_{r^{+}}(3,k,n)).\]  
Theorem 3.1: _Let \(k\geq 1,\ k\
eq 3\), \(n\geq 1\) be integers and \(r=min\{3,k\}-1\). Then_  
1. \(\gamma_{0}(3,k,n)=F_{3}^{(1)}(k,n),\)__
2. \(\gamma_{\leq r^{-}}(3,k,n)=\gamma_{\leq r^{+}}(3,k,n)=F_{3}^{(2)}(k,n),\)__
3. \(\gamma_{\leq r^{-}}(3,k,n)-\gamma_{0}(3,k,n)=F_{3}^{(3)}(k,n),\)__
4. \(\gamma(3,k,n)=F_{3}^{(1)}(k,n)+2F_{3}^{(2)}(k,n).\)__  
Proof. : (i) (by induction on \(n\)) At the beginning, we prove that the equality (i) holds for \(n\in\{1,\ldots,max\{3,k\}-1\}\). If \(k=1\), then \(n\in\{1,2\}\), and consequently we obtain the unique \((3,1)\)-decomposition of \(n\) of the form \((1)\) or \((1,1)\), respectively. Thus, \(\gamma_{0}(3,1,1)=F_{3}^{(1)}(1,1)\), and \(\gamma_{0}(3,1,2)=F_{3}^{(1)}(1,2)\). If \(k=2\), then there is no \((3,2)\)-decomposition of \(n=1\) and there is the unique \((3,2)\)-decomposition of \(n=2\) of the form \((2)\), so \(\gamma_{0}(3,2,1)=F_{3}^{(1)}(2,1)\), and \(\gamma_{0}(3,1,2)=F_{3}^{(1)}(2,2)\). Assume now that \(k>3\). If \(1\leq n\leq\max\{3,k\}-1\), then \(\gamma_{0}(3,k,n)=1\) if \(n=3p\), \(p\geq 1\), otherwise \(\gamma_{0}(3,k,n)=0\). Hence, \(\gamma_{0}(3,k,n)=F_{3}^{(1)}(k,n)\) for initial values \(1\leq n\leq\max\{3,k\}-1\). Let \(n\geq\max\{3,k\}\) and suppose that \((i)\) is true for an arbitrary \(m<n\). We shall show that \(\gamma_{0}(3,k,n)=F_{3}^{(1)}(k,n)\). Let us consider a \(t\)-tuple \((n_{1}\ldots,n_{t})\), \(t\geq 1\) being a \((3,k)\)-decomposition of \(n\). If \(n_{t}=k\), then \(n=\sum\limits_{i=1}^{t-1}n_{i}+k\), and so \(n-k=\sum\limits_{i=1}^{t-1}n_{i}\). By induction hypothesis there are \(F_{3}^{(1)}(k,n-k)\) distinct \((3,k)\)-decompositions of \(n\) with \(n_{t}=k\).