A certain problem in equal sums of like powers for $7$th powers entails the elliptic curve,
$$u(u+127^2)(u+129^2) = y^2$$
I was looking at the general case,
$$u \big(u + (n - 1)^2\big) \big(u + (n + 1)^2\big) = y_1^2\tag1$$
and using this online Magma to test integer $n>1$ such that it has positive rank. It starts as,
$$n = 6, 13, 16, 18, 22, 23, 32, 33, 35, 36, 37, 41, 42, 43, 44, 45, 46, 50,\dots$$
Checking the OEIS, it turns out it may be sequence A228380 but involves the $n$ of the elliptic curve,
$$v(v - 1)(v - n^2) = y_2^2\tag2$$
such that it has positive rank.
Q: Is it true that the sequence of $n>1$ for $(1)$ and $(2)$ are in fact the same? If so, why?