# New Types of Distance Padovan Sequences via Decomposition Technique
## 1 Introduction and Preliminary Results  
Let \(k\geq 1\) be an integer. Sequences defined by the \(k\)-th order linear homogeneous recurrence equation of the form  
\[a_{n}=b_{1}a_{n-1}+b_{2}a_{n-2}+\cdots+b_{k}a_{n-k},\ \ \text{for}\ \ n\geq k, \tag{1}\]  
where coefficients \(b_{j}\geq 0\) for \(j\in\{1,\ldots,k-1\}\) and \(b_{k}>0\) are integers, with fixed integers \(a_{i},i\in\{0,\ldots,k-1\}\), are called Fibonacci-type (or Fibonacci-like) sequences, and their elements are called numbers of the Fibonacci type (or Fibonacci-like), respectively. We denote by \(\mathcal{F}\) the family of Fibonacci-type sequences defined by the Equation (1) and by \(F(a_{i};b_{j})\) for \(i\in\{0,\ldots,k-1\}\) and \(j\in\{1,\ldots,k\}\) the sequences belonging to \(\mathcal{F}\). The family \(\mathcal{F}\) includes a number of well-known sequences that have been intensively studied in the literature. Below, we list some classical Fibonacci-type sequences defined by the second- and the third-order linear recursions. The identification number of each sequence coming from the On-line Encyclopedia of Integer Sequences, which is an online database of integer sequences [1], is given in brackets. \(F(1,1;1,1)\)--Fibonacci sequence (A000045). \(F(2,1;1,1)\)--[PERSON] sequence (A000032). \(F(0,1;2,1)\)--[PERSON] sequence (A000129). \(F(2,2;2,1)\)--[PERSON] sequence (A0002203). \(F(0,1;1,2)\)--[PERSON] sequence (A001045). \(F(2,1;1,2)\)--[PERSON] sequence (A014551). \(F(1,1,1;1,0,1)\)--[PERSON] sequence (A000930). \(F(1,1,1;0,1)\)--[PERSON] sequence (A000931). \(F(3,0,2;0,1,1)\)--[PERSON] sequence (A001608). For other well-known sequences from \(\mathcal{F}\) defined by the second- or the third-order recurrence equation, see, for example, [2]. Sequences from the family \(\mathcal{F}\) are widely studied in many areas of modern research from different points of view, for example, in medicine [3],theoretical physics [4], particularly physics of the high energy [5; 6; 7], engineering, computer science [8; 9], architecture, nature, arts, [10; 11], and others. For historical background issues of the particular cases of Fibonacci-type sequences, see [12]. The family \(\mathcal{F}\) includes a special subfamily of distance Fibonacci-type sequences of the form \(F(a_{i};b_{j})\), where \(i\in\{1,\ldots,k-1\},j\in\{1,\ldots,k\}\), \(b_{k}\
eq 0\), \(b_{p}\
eq 0\) for a fixed \(p\) such that \(1\leq p\leq k-1\), and \(b_{j}=0\) for \(j\in\{1,\ldots,k\}\setminus\{p,k\}\). In other words, the subfamily of distance Fibonacci-type sequences consists of sequences that are defined by the \(k\)-th-order recurrence equation, and each term of such a sequence is the sum of two terms, such that one of them is at the distance \(k\) from \(n\), for an arbitrary \(k\geq 2\). For convenience, we will write \(F(a_{i};b_{p}=n_{p},b_{k}=n_{k})\) for this subfamily. In this subfamily, the most interesting are sequences that generalize the well-known classical sequences. These sequences are intensively studied in different contexts, for example, for their combinatorial properties, graph interpretations, or hypercomplex number theory. Below, we recall some distance Fibonacci-type sequences related to the main topic of this paper. \(F(a_{i}=i+1;b_{1}=b_{k}=1)\), \(i\in\{0,\ldots,k-1\}\), \(k\geq 2\) ([PERSON] et al. [13]). \(F(a_{i}=1;b_{k-1}=b_{k}=1)\), \(i\in\{0,\ldots,k-1\}\), \(k\geq 2\) ([PERSON] et al. [14]). \(F(a_{i}=1;b_{2}=b_{k}=1)\), \(i\in\{0,\ldots,k-1\}\), \(k\geq 1\) ([PERSON] et al. [15]). \(F(a_{i}=1;b_{3}=b_{k}=1)\), \(i\in\{0,\ldots,k-1\}\), \(k\geq 1\) ([PERSON] et al. [16]). The above sequences generalize Fibonacci sequence, [PERSON] sequence, Padovan sequence, and others. In this paper, being a sequel of the paper [16], we introduce a subfamily \(F(a_{i};b_{3}=b_{k}=1)\), where \(i\in\{0,\ldots k-1\}\), generated by special decompositions of a number \(n\). Sequences of this subfamily generalize the classical Padovan sequence \(F(1,1,1;0,1,1)\) and Narayana sequence \(F(1,1,1;1,0,1)\) simultaneously.