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This Demonstration displays integer sequences of the form

and

, where

and

are the roots of the quadratic equation

with integer coefficients

and

and

are integers, then the roots of the quadratic equation

are

and

It turns out that

produces a Lucas-like sequence, while

produces a Fibonacci-like sequence. In particular, if

, we obtain the Lucas and Fibonacci numbers.

Reference:

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A Family of Generalized Fibonacci and Lucas Numbers