Parity

The parity of an integer is its attribute of being even or odd. Thus, it can be said that 6 and 14 have the same parity (since both are even), whereas 7 and 12 have opposite parity (since 7 is odd and 12 is even).

The binary parity of a nonnegative integer is the remainder of its binary digit sum on division by 2. The sum of the bits in its binary representation is the digit count . For example, has two 1s in its binary representation and hence has binary parity 0. The binary parities of the first few nonnegative integers are therefore 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, ... (OEIS A010060), as summarized in the following table.

	binary	parity		binary	parity
1	1	1	11	1011	1
2	10	1	12	1100	0
3	11	0	13	1101	1
4	100	1	14	1110	1
5	101	0	15	1111	0
6	110	0	16	10000	1
7	111	1	17	10001	0
8	1000	1	18	10010	0
9	1001	0	19	10011	1
10	1010	0	20	10100	0

A generating function for binary parity is given by

(1)

for . The constant generated by interpreting the sequence of binary parity digits as a binary fraction is called the Thue-Morse constant.

The binary parity function obeys the sum identity

(2)

for any nonnegative integer and any complex number . For example, for and ,

(3)