The infinite lower triangular matrix $\mathbb{B}(r_1,dots,r_l;s_1,dots,s_{l'})$ is considered over the sequence space $c_0$, where $l$ and $l'$ are positive integers. The diagonal and sub-diagonal entries of the matrix consist of the oscillatory sequences $r=(r_{k (\text{mod} \ l)+1})$ and $s= (s_{k(\text{mod} \ l')+1})$, respectively. The rest of the entries of the matrix are zero. It is shown that the matrix represents a bounded linear operator. Then the spectrum of the matrix is evaluated and partitioned into its fine structures: point spectrum, continuous spectrum, residual spectrum, etc. In particular, the spectra of the matrix $\mathbb{B}(r_1,dots, r_4; s_1,dots, s_{6})$ are determined. Finally, an example is taken in support of the results.