Number of numbers between n-1 and n that are the sum of consecutive squares.

0, 0, 1, 2, 0, 2, 1, 4, 1, 4, 2, 4, 2, 4, 4, 3, 5, 3, 2, 6, 3, 4, 6, 3, 5, 5, 4, 5, 6, 4, 4, 9, 2, 6, 5, 8, 3, 4, 10, 3, 5, 6, 7, 5, 5, 11, 2, 7, 5, 10, 5, 6, 5, 11, 3, 9, 3, 11, 6, 4, 7, 10, 6, 6, 5, 13, 4, 7, 4, 13, 3, 7, 7, 6, 11, 6, 5, 6, 14, 6, 6, 7, 9, 10

1

Surprizingly irregular! However, at higher powers, the plot becomes even more complex.

T. D. Noe, Plot of 10000 terms

T. D. Noe, Table of 10000 terms

(Mma) nMax = 100; pwr = 2; t = Reap[Do[k = n; s = 0; While[s = s + k^pwr; s <= nMax^pwr, Sow[s]; k++], {n, nMax}]]; t = Union[t[[2, 1]]]; t = Complement[t, Range[0, nMax]^pwr]; ta = Tally[Floor[t^(1/pwr)]]; tb = Complement[Range[0, nMax - 1], Transpose[ta][[1]]]; tc = Transpose[{tb, Table[0, {Length[tb]}]}]; Transpose[Union[ta, tc]][[2]]

nonn

T. D. Noe, May 14 2015