There appears to be no mention in OEIS:  [Quarter-squares, A002620][1].  Can someone give a proof or reference?

Examples:

quarter-squares: ${0,1,2,4,6,9,12,16,20,25,30,36,...}$

2-term sums: ${2+1, 4+1, 6+2, 6+2,...,90+9,...}$

3-term sums: ${12+2+1, 16+2+1,...,72+6+2,...}$

4-term sums: ${240+12+2+1,...,6480+72+6+2,...}$



[https://oeis.org/A002620][2].