We can study properties shared by Fibonacci and Lucas numbers by investigating a number sequence that satisfies the Fibonacci recurrence, but with arbitrary initial conditions. This is our focus in this chapter.
7.1 GIBONACCI NUMBERS
Consider the sequence , where , and . The ensuing sequence
is the gibonacci sequence (generalized Fibonacci sequence); is the nth gibonacci number. (A.T. Benjamin and J.J. Quinn coined the term gibonacci .)
Take a good look at the coefficients of a and b in the various terms of this sequence. They follow the same pattern we saw in Chapter 5 for powers of : The coefficients of a and b are Fibonacci numbers. We will now pinpoint these two coefficients in the following theorem.