Perfect square sequence In base 10, the sequence 49,4489,444889,... consists of all perfect squares.
Is this true for any other bases (greater than 10, of course)?

$((2*19^5+1)/3^2)=2724919437289,\ \ $ which converted to base $19$ is $88888GGGGH$.  It doesn't work in base $13$ or $16$.  In base $28$ it gives $CCCCCOOOOP$, where those are capital oh's (worth $24$).
This is because if we express $\frac{1}{9}$ in base $9a+1$, it is $0.aaaa\ldots$.  So $\left (\frac{2(9a+1)^5+1}{3}\right)^2=\frac{4(9a+1)^10+4(9a+1)^5+1}{9}=$
$ (4a)(4a)(4a)(4a)(4a)(4a)(8a)(8a)(8a)(8a)(8a+1)_{9a+1}$ 
where the parentheses represent a single digit and changing the exponent from $5$ changes the length of the strings in the obvious way.
