On a new improved unifying closed formula for all Fibonacci-type sequences and some applications

In this paper, we establish a new general framework under which all generalized Fibonacci-type sequences come together. The main result lies in obtaining one new general closed formula solution for all such sequences by using only matrix theory. This new formula in turn gives new closed formula solutions for most well-known sequences of this type. Next, the advantages of this formula are exploited. In particular, two applications are presented; one deals with finding the most general congruence formula for such sequences which generalizes most known formulas of this type and the other is concerned with computational number theory.