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.


Consider the sequence c07-math-001, where c07-math-002, and c07-math-003. The ensuing sequence


is the gibonacci sequence (generalized Fibonacci sequence); c07-math-004 is the nth gibonacci number. (A.T. Benjamin and J.J. Quinn coined the term gibonacci [35].)

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 c07-math-005: The coefficients of a and b are Fibonacci numbers. We will now pinpoint these two coefficients in the following theorem.

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