elementary number theory - My formula for sum of consecutive squares series?

I stumbled upon a specific series, who's Sum of squares of consecutive integers equals the sum of squares of the continuation of that consecutive integers.

For exmaple, this first number in the series results in: $3^2 + 4^2 = 5^2$.

The second results in: $10^2 + 11^2 + 12^2 = 13^2 + 14^2$.

The thrid: $21^2 + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2$. etc.

The formula for the series is:

By using; $1^2 + 2^2 + ... + n^2 = \frac{n(n+1)(2n+1)}{6}$ it is easy to prove that both side are the same.

Have I found something new? Useful?

I have a feeling that this is the only series with this specific property (consecutive sum of squares equal continuation of consecutive sum of squares), but I am unsure how to prove that.

Any comments would be appreciated.