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

It is probably coincidence. When considering sums of 16 or less consecutive squares, where the smallest is at most $200^2$, I found only one other such sum:

$$71^2+72^2+73^2+74^2+75^2+76^2+77^2+78^2=44444$$

rare. Seems to be impossible for numeral bases $5, 11.$ I reported the numbers in base ten, you need to imagine how to write them in the specified base. Base 15 does not seem to work, except for 13; that is, as the base gets larger, we get sums of consecutive squares that are smaller than the base and thus require just one digit. Base 16 ("hexadecimal") does well, we get $1111111_{16}.$

 base   17
54               2   5
90               2   6
126               4   7
2456               4   19
 end base   17

 base   16
85               6   7
204               1   8
255               5   9
221               10   11
819               1   13
2730               8   20
17895697               624   666
 end base   16

 base   14
30               1   4
90               2   6
135               3   7
20685               25   42
 end base   14

 base   13
14               1   3
140               1   7
126               4   7
366               8   11
4760               8   24
4760               23   29
11900               18   34
23800               18   42
11900               35   42
 end base   13

 base   12
13               2   3
91               1   6
7540               15   29
9425               5   30
 end base   12

 base   10
55               1   5
77               4   6
1111               11   16
44444               71   78
444444               51   113
 end base   10

 base   9
30               1   4
50               3   5
91               1   6
728               7   13
44286               16   51
332150               53   104
 end base   9

 base   8
54               2   5
365               10   12
365               13   14
1755               5   17
3510               10   22
 end base   8

 base   7
285               1   9
 end base   7

 base   6
14               1   3
86               3   6
1036               9   15
9331               19   32
 end base   6

 base   4
5               1   2
85               6   7
255               5   9
2730               8   20
 end base   4

 base   3
13               2   3
728               7   13
 end base   3

 base   2
255               5   9
 end base   2

This is the meat of the program, C++ with GMP

mpz_class choose3(mpz_class n)
{
  return n * (n-1) * (n-2) / 6;
}


mpz_class squarepyramid(mpz_class n)
{
  return choose3(n+2 ) + choose3(n+1 ) ;
}


int main()
{
  mpz_class base = 10;
  cout << " base   " << base << endl;
  for(mpz_class x = 2; x <= 12200; ++x) {
   for(mpz_class y = 0; y < x -1; ++y) {

 // cout << x << "   " << squarepyramid(x) << endl;
   mpz_class m = squarepyramid(x) - squarepyramid(y) ;
   mpz_class n =  1;
   while (n < m) n *= base;
   n -= 1;
   n /= (base - 1);
   if( m % n == 0 && m >= base) cout << m << "               "  << y+1 <<  "   "  << x << endl;

  }}
  cout << " end base   " << base << endl;
    return 0 ;
}