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Certain Matrices Associated with Balancing and Lucas-balancing Numbers
| Content Provider | Semantic Scholar |
|---|---|
| Author | Ray, Prasanta Kumar |
| Copyright Year | 2012 |
| Abstract | \cdots Balancing numbers $n$ and balancers $r$ are originally defined as the solution of the Diophantine equation $1+2+\cdots+(n-1)=(n+1)+(n+2)+\cdots+(n+r)$. These numbers can be generated by the linear recurrence $B_{n+1}=6B_{n}-B_{n-1}$ or by the nonlinear recurrence $B_{n}^{2}=1+B_{n-1} B_{n+1}$. There is another way to generated balancing numbers using powers of a matrix $Q_{B} = \begin{pmatrix} 6 & -1 \\ 1 & 0\\ \end{pmatrix}.$ The matrix representation, indeed gives many known and new formulas for balancing numbers. In this paper, using matrix algebra we obtain several interesting results on balancing and related numbers. Keywords: Balancing numbers; Lucas-balancing numbers; Triangular numbers; Recurrence relation; Balancing Q-matrix; Balancing R-matrix. 2010 Mathematics Subject Classification: 11B39, 11B83 |
| Starting Page | 15 |
| Ending Page | 22 |
| Page Count | 8 |
| File Format | PDF HTM / HTML |
| DOI | 10.11113/matematika.v28.n.311 |
| Volume Number | 28 |
| Alternate Webpage(s) | https://matematika.utm.my/index.php/matematika/article/download/311/304 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
