MathDB

We consider all partitions of a positive integer n into a sum of (nonnegative integer) exponents of

2

(i.e.

1

2

4

8

. . .

). A number in the sum is allowed to repeat an arbitrary number of times (e.g.

7 = 2 + 2 + 1 + 1 + 1

) and two partitions differing only in the order of summands are considered to be equal (e.g.

8 = 4 + 2 + 1 + 1

and

8 = 1 + 2 + 1 + 4

are regarded to be the same partition). Let

E(n)

be the number of partitions in which an even number of exponents appear an odd number of times and

O(n)

the number of partitions in which an odd number of exponents appear an odd number of times. For example, for

n = 5

partitions counted in

E(n)

are

5 = 4 + 1

and

5 = 2 + 1 + 1 + 1

, whereas partitions counted in O(n) are

5 = 2 + 2 + 1

and

5 = 1 + 1 + 1 + 1 + 1

, hence

E(5) = O(5) = 2

. Find

E(n) - O(n)

as a function of

n

Problem Statement