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 $P=(x_1\mathrm{/}{z}_1^2,y_1\mathrm{/}{z}_1^3)$ be a rational point of infinite order on $E$, where $x_1,y_1,z_1$ are coprime integers. We show that the integer sequence $(z_n)$ defined by $nP=(x_n\mathrm{/}{z}_n^2,y_n\mathrm{/}{z}_n^3)$ for all $n\ge 1$ does not eventually coincide with $(u_{n^2})$ for any choice of linear recurrence sequence $(u_n)$ with integer values.