What stands out in this article is the sequence spaces of a new brand c(0)(lambda)((B) over tilde) and c(lambda)((B) over tilde), derived by using a double sequential band matrix B(T-,'s-) which generalizes the previous work of Sonmez and Baar (Abstr. Appl. Anal. 2012:435076,2012), where (r(n))(n=0)(infinity) and (s(n))(n=0)(infinity) are given convergent sequences of positive real numbers. The aforementioned spaces are in fact the BK-spaces of non-absolute type. Moreover, they are norm isomorphic to the spaces co and c, respectively. Then, some inclusion relations are derived to determine the alpha-, beta- and gamma-duals of these spaces. Next, their Schauder bases are constructed. In conclusion, some matrix classes from the spaces c(0)(lambda)((B) over tilde) and c(lambda)((B) over tilde) to the spaces l(p), C-o and c are characterized. When compared with the corresponding results in the literature, it is seen that the results of the present study are more general and more inclusive.