Generalizations of Fibonacci Numbers - Fibonacci Word

Fibonacci Word

In analogy to its numerical counterpart, the Fibonacci word is defined by:

F_n := F(n):= \begin{cases} b & \mbox{if } n = 0; \\ a & \mbox{if } n = 1; \\ F(n-1)+F(n-2) & \mbox{if } n > 1. \\ \end{cases}

where + denotes the concatenation of two strings. The sequence of Fibonacci strings starts:

b, a, ab, aba, abaab, abaababa, abaababaabaab, …

The length of each Fibonacci string is a Fibonacci number, and similarly there exists a corresponding Fibonacci string for each Fibonacci number.

Fibonacci strings appear as inputs for the worst case in some computer algorithms.

If "a" and "b" represent two different materials or atomic bond lengths, the structure corresponding to a Fibonacci string is a Fibonacci quasicrystal, an aperiodic quasicrystal structure with unusual spectral properties.

Read more about this topic:  Generalizations Of Fibonacci Numbers

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