The Metallic Ratio of Pulsating Fibonacci Sequences

The golden ratio and the Fibonacci sequence (Fn) are well known, as is the fact that the ratio Fn+1Fn converges to the golden ratio for sufficiently large n. In this paper, we investigate the metallic ratio—a generalized version of the golden ratio—of pulsating Fibonacci sequences in three forms. Two of these forms are considered in the sense of pulsating recurrence relations, and their diagrams can be represented by symmetry, which is one of their distinguishing characteristics. The third form is the Fibonacci sequence in bipolar quantum linear algebra (BQLA), which also pulsates.