What's the name for sequences sitting between geometric and arithmetic, i.e. whose recurrence relation is of the form $ax+b$?

Question

What's the name for sequences sitting between geometric and arithmetic?

E.g. let $x_{n+1}=ax_n+b$

I can't find a general name for these.

These sequences may also be a Lucas Sequence but that's a pretty convoluted way of defining these.

Answer 1

I think the term you are looking for is Linear recurrence Relations

Answer 2

In a french context they are called "suites arithmético-géométriques".

Answer 3

The relations $x_{n+1}=ax_n+b\,$ ($n=0,1,...$), with $a$ and $b$ constant, describe a form of iterated affine transformation, and so the generated $x_0,x_1,...$ could be called an affine sequence. The iterative process has been called an affine recursion. You can do an internet search for this term, but I only found more advanced applications of this idea, to generate fractals or solve minimization problems in higher dimensions.