An elliptic sequence is not a sampled linear recurrence sequence

Let E be an elliptic curve defined over the rationals and in minimal Weierstrass form, and let P= (x₁/z₁², y₁/z₁³) be a rational point of infinite order on E, where x₁, y₁, z₁ are coprime integers. We show that the integer sequence (zn) n≥1 defined by nP= (xn/z²n, yn/z³n) for all n≥1 does not eventually coincide with (un²) n≥1 for any choice of linear recurrence sequence, (un)n≥1 with integer values.