Okay, so it's not quite a duplicate because I guess you're asking about initial conditions as well. The generating functions of the sequences $a_n$ which have this property are called $\mathbb{N}$-recognizable or $\mathbb{N}$-rational in the literature, and they include the generating functions of word lengths in regular languages. Not all rational functions with non-negative integer coefficients are $\mathbb{N}$-rational; see for example the counterexamples here.

Stanley's Enumerative Combinatorics discusses some of these issues, in particular look at Section 4.7.

Okay, so it's not quite a duplicate because I guess you're asking about initial conditions as well. The generating functions of the sequences $a_n$ which have this property are called $\mathbb{N}$-recognizable or $\mathbb{N}$-rational in the literature, and they are essentially (precisely?) the generating functions of word lengths in regular languages (star example: the look-and-say sequence). Not all rational functions with non-negative integer coefficients are $\mathbb{N}$-rational; see for example the counterexamples in these slides. These slides also seem relevant.

Stanley's Enumerative Combinatorics discusses some of these issues, in particular look at Section 4.7.