The resurgence of an ideal of fat points in đ[âN] with linear minimal free resolution can be expressed as the quotient of its initial degree and its Waldschmidt constant. This makes it possible to do computations of the resurgence of this type of ideal with less complexity than in general. In this paper, given a fat point subscheme of âN, we construct a new subscheme where its saturated ideal has a linear minimal free resolution. In particular, we show that the saturated ideal of a fat point subscheme Z=(sâ2)p1+p2+âŻ+ps, supported on sâ„3 general points of â2, has a linear minimal free resolution, and we compute its resurgence.