sum of binary digit representation of n, mod 2

0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, ...

repeatedly apply the replacement rule {0->01, 1->10} on 0

a_n = Mod[Total[IntegerDigits[n, 2]], 2]

Table[Mod[Total[IntegerDigits[k, 2]], 2], {k, 0, n}]