  In literature, there are two different definitions of elliptic divisibility
sequences. The first one says that a sequence of integers $\{h_n\}_{n\geq 0}$
is an elliptic divisibility sequence if it verifies the recurrence relation
$h_{m+n}h_{m-n}h_{r}^2=h_{m+r}h_{m-r}h_{n}^2-h_{n+r}h_{n-r}h_{m}^2$ for every
natural number $m\geq n\geq r$. The second definition says that a sequence of
integers $\{\beta_n\}_{n\geq 0}$ is an elliptic divisibility sequence if it is
the sequence of the square roots (chosen with an appropriate sign) of the
denominators of the abscissas of the iterates of a point on a rational elliptic
curve. It is well-known that the two sequences are not equivalent. Hence, given
a sequence of the denominators $\{\beta_n\}_{n\geq 0}$, in general does not
hold
$\beta_{m+n}\beta_{m-n}\beta_{r}^2=\beta_{m+r}\beta_{m-r}\beta_{n}^2-\beta_{n+r}\beta_{n-r}\beta_{m}^2$
for $m\geq n\geq r$. We will prove that the recurrence relation above holds for
$\{\beta_n\}_{n\geq 0}$ under some conditions on the indexes $m$, $n$, and $r$.
