The sequence of Lucas polynomials is defined by , , and for

The next few terms in the sequence are , and .

By induction, the degree of is equal to . The integers are the Lucas numbers, which are natural “companions” to the Fibonacci numbers (see, e.g., this blog post).

The polynomials

It’s easy to see that for odd, is divisible by and has only even-power terms. Thus for some monic integer polynomial of degree . We will be particularly interested in the polynomials for prime.

If is even (resp. odd), a simple induction shows that the constant term (resp. the coefficient of ) in is equal to . In particular, for odd we have .