POWER FIBONACCI SEQUENCES

We examine integer sequences G satisfying the Fibonacci recurrence relation G(n) = G(n-1)+G(n-2) that also have the property G equivalent to 1, a, a(2), a(3),... (mod m) for some modulus m. We determine those moduli m for which these power Fibonacci sequences exist and the number of such sequences for a given m. We also provide formulas for the periods of these sequences, based on the period of the Fibonacci sequence F modulo m. Finally, we establish certain sequence/subsequence relationships between power Fibonacci sequences.