A037845 Sum{d(i)-d(i-1): d(i)>d(i-1), i=0,1,...,m}, where Sum{d(i)*4^i: i=0,1,...,m} is base 4 representation of n.

0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 2, 3, 0, 0, 1, 2, 1, 1, 1, 2, 2, 2, 2, 2, 0, 1, 2, 3, 0, 0, 1, 2, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 2, 3, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 2, 3, 1, 1, 2, 3, 2, 2, 2, 3, 3, 3, 3, 3, 0, 1, 2, 3, 0, 0, 1, 2, 1, 1, 1