An angel at my picnic

The ruler function: 2^a(n) divides 2n. Or, a(n) = 2-adic valuation of 2n.  ( OEIS A001511 )
1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, ...
Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4).   ( OEIS A002620 )
0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90, 100, 110, 121, 132, 144, 156, 169, 182, 196, 210, 225, 240, 256, ...
Some examples :
for n=2, 3, 4, 5, 6, 7, 8, 9, 10  have  1*1,  1*2,  2*2,  2*3,  3*3,  3*4,  4*4,  4*5  and  5*5