The first example in recmath.org/PolyPages/PolyPages/index.htm?RepO.htm is a hexomino (two rows, aligned to the left, with 2 boxes in one of the rows and 4 rows in the other) that is reptile for all n.

As far as I can tell math.ntnu.no/~stacey is not available anymore. I failed to find it elsewhere. If you have time, could you point me to it? I'm mostly interested in the integration with daily arXiv postings.

Never mind, sorry, I wasn't searching for the right keyword: "Why are local systems and representations of the fundamental group equivalent" and "Why are local systems on a complex analytic space equivalent to vector bundles with flat connection?" seem to answer me.

@diverietti Do you have by any chance a reference on the well-known "equivalence between the isomorphism classes of pairs (E,∇), where E→M is a complex vector bundle of rank r and ∇ a flat connection on E, and the isomorphism classes of complex linear representations of dimension r of the fundamental group π₁(M) of M."?

Probably irrelevant comment: twice the first expression, minus the second, gives that $(p_1^2 - 4p_2) [M] \equiv 0 \pmod 3$.