Parity

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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).A different type of parity of an integer is defined as the sum of the bits in binary representation, i.e., the digit count , computed modulo 2. So, for example, the number has two 1s in its binary representation and hence has parity 2 (mod 2), or 0. The parities of the first few integers (starting with 0) are therefore 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, ... (OEIS A010060), as summarized in the following table.binaryparitybinaryparity11111101112101121100031101311011410011411101510101511110611001610000171111171000108100011810010091001019100111101010020101000A generating function for parity is givenby(1)The constant generated by interpreting the sequence of parity digits as a binary fraction is called the Thue-Morse constant.The parity..

Binary carry sequence

The sequence given by the exponents of the highest power of 2 dividing , i.e., the number of trailing 0s in the binary representation of . For , 2, ..., the first few are 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, ... (OEIS A007814).Amazingly, this corresponds to one less than the number of disks to be moved at th step in the optimal solution to the tower of Hanoi problem: 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, ... (OEIS A001511). The parity of this sequence is given by 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, ... (OEIS A035263) which, amazingly, also corresponds to the accumulation point of cycles through successive bifurcations.

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