For fixed integer parameters $(P,Q)$, [Lucas sequences](https://en.wikipedia.org/wiki/Lucas_sequence) represent a pair of complimentary integer sequences satisfying the same recurrence with the characteristic polynomial $f(x):=x^2 - Px + Q$. The two sequences can be defined via roots $x_{1,2}$ of $f(x)$ as $U_n:=\frac{x_1^n - x_2^n}{x_1-x_2}$ and $V_n:=x_1^n + x_2^n$.

Lucas sequences form a linear basis for the other sequences satisfying the same recurrence, and are tightly interconnected with [numerous identities](https://en.wikipedia.org/wiki/Lucas_sequence#Other_relations) connecting arithmetic operations on terms to those on their indices.

> Q: What would be a generalization of Lucas sequences to integer recurrences of higher order, in particular of order 3 with a characteristic polynomial $g(x):=x^3 - Px^2 + Qx - R$ with integer coefficients?

Generalized Lucas sequences should be integer, form a basis for the other sequences with the same characteristic polynomial, and have nice interconnecting properties. I assume one of the sequences should still be the [power sum symmetric polynomial](https://en.wikipedia.org/wiki/Power_sum_symmetric_polynomial) of the roots of $g(x)$, but what would be the others?