sums of distinct powers of 4

0, 1, 4, 5, 16, 17, 20, 21, 64, 65, 68, 69, 80, 81, 84, 85, 256, 257, 260, 261, 272, ...

a_n = FromDigits[Riffle[IntegerDigits[n, 2], 0], 2]

Table[FromDigits[Riffle[IntegerDigits[k, 2], 0], 2], {k, 0, n}]