Is this (deceptive) pattern just a coincidence?

At the moment it seems that this is an "assisted coincidence". Thanks to Thomas Andrews I know that the expressed quantities are congruent to $6$ mod $60$, and this reduces significantly the possible results, so that numbers like $66$ and $666$ no longer seem unbelievably lucky.

Thanks to Greg Martin it is easy to understand why the pattern couldn't last forever. I'll write the would-be general term of the pattern $$ 1^{2n}-2^{2n}+3^{2n} \stackrel{?}{=} 6\cdot\sum_{k=0}^{n-1}10^k $$ Dividing by $10^n$ one can see that the right-hand side tends towards $2/3$: $$ \lim_{n\rightarrow +\infty}\frac{6}{10^n}\cdot\sum_{k=0}^{n-1}10^k = 6\cdot\lim_{n\rightarrow +\infty}\sum_{k=0}^{n-1}\left(\frac{1}{10}\right)^{n-k}= 6\cdot\lim_{n\rightarrow +\infty}\sum_{d=1}^{n}\left(\frac{1}{10}\right)^d=6\cdot\frac{1}{9}=\frac{2}{3} $$ while the left-hand side tends towards $0$ ( $\sim(3^2/10)^n$ ), having a chance to be equal to the right side only for little $n$'s. I've made a plot to show this, switching to $x$, where one can see the fortunate interceptions at $(1,0.6)$, $(2,0.66)$ and $(3,0.666)$, while after those the curves separate definitively. See images below.

I love Peter's remark that $6$, $66$ and $666$ are triangular numbers, while $6666$ is not, but I don't see a link to the original question.