Collection from a sequence that constitute a perfect square
05AB1E, 10 bytes
Ý¨n6*æʒOÅ²

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Ý¨n6*æʒOÅ² || Full program. I'll call the input N.

Ý          || 0-based inclusive range. Push [0, N] ∩ ℤ.
 ¨         || Remove the last element.
  n        || Square (element-wise).
   6*      || Multiply by 6.
     æ     || Powerset.
      ʒ    || Filter-keep those which satisfy the following:
       O   ||---| Their sum...
        Å² ||---| ... Is a perfect square?