<s>[INST] Is $11^2+12^2+13^2+14^2+15^2+16^2=1111$ special? Is this pure coincidence or is this a special case of some well-known number-theoretic result?
If the latter is true, is there some notable generalization?
EDIT: Thanks to the interesting answers below, a follow-up question is now on MathOverflow: https://mathoverflow.net/questions/239033/repdigit-numbers-which-are-sum-of-consecutive-squares?noredirect=1#comment591447_239033
 [/INST] The reason is that $$11^2+12^2+13^2+14^2+15^2+16^2=600+420+91$$ in which the summands are such that successively give $1,1,1,1$  when making the sum. In fact,
$$(10+1)^2+(10+2)^2+(10+3)^2+(10+4)^2+(10+5)^2+(10+6)^2=6\cdot10^2+20(1+2+3+4+5+6)+(1^2+2^2+3^2+4^2+5^2+6^2)=6\cdot10^2+20(\frac{6\cdot7}{2})+\frac{6(6+1)(2\cdot6+1)}{6}=600+420+91$$
I don't know if it is generalizable.
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