Sequence of Squares Beginning and Ending with n 4s

Sequence
The following sequence:

cannot be continued, as it is not possible for there to be a square number ending in $\ldots 4444$.

Proof
Let $n = 10000 k + 4444$.

We have:

By Square Modulo 4, $\dfrac n 4$ cannot be a square number.

Therefore neither can $n = 4 \times \dfrac n 4$ be a square number.