Numbers which are not the sum of distinct squares

Integer squares are the set of integers multiplied by themselves: 1 x 1 = 1, 2 × 2 = 4, 3 × 3 = 9, etc. ( 1, 4, 9, 16 ... )

Most positive integers can be generated as the sum of 1 or more distinct integer squares.

     1 == 1
     5 == 4 + 1
    25 == 16 + 9
    77 == 36 + 25 + 16
   103 == 49 + 25 + 16 + 9 + 4

Many can be generated in multiple ways:

    90 == 36 + 25 + 16 + 9 + 4 == 64 + 16 + 9 + 1 == 49 + 25 + 16 == 64 + 25 + 1 == 81 + 9
   130 == 64 + 36 + 16 + 9 + 4 + 1 == 49 + 36 + 25 + 16 + 4 == 100 + 16 + 9 + 4 + 1 == 81 + 36 + 9 + 4 == 64 + 49 + 16 + 1 == 100 + 25 + 4 + 1 == 81 + 49 == 121 + 9

A finite number can not be generated by any combination of distinct squares:

   2, 3, 6, 7, etc.

Task

Find and show here, on this page, every positive integer than can not be generated as the sum of distinct squares.

Do not use magic numbers or pre-determined limits. Justify your answer mathematically.

See also

OEIS: A001422 Numbers which are not the sum of distinct squares

Raku

Try it online!

Spoiler: (highlight to read)
Once the longest run of consecutive generated sums is longer the the next square, every number after can be generated by adding the next square to every number in the run. Find the new longest run, add the next square, etc. <lang perl6>my @squares = ^∞ .map: *²; # Infinite series of squares

for 1..∞ -> $sq { # for every combination of all squares

   my @sums = @squares[^$sq].combinations».sum.unique.sort;
   my @run;
   for @sums {
       @run.push($_) and next unless @run.elems;
       if $_ == @run.tail + 1 { @run.push: $_ } else { last if @run.elems > @squares[$sq]; @run = () }
   }
   put grep * ∉ @sums, 1..@run.tail and last if @run.elems > @squares[$sq];

}</lang>

Output:

2 3 6 7 8 11 12 15 18 19 22 23 24 27 28 31 32 33 43 44 47 48 60 67 72 76 92 96 108 112 128