For a given positive integer, we want to partition it into the sum of

some positive integers, or itself. For example, 1 to 4 can be

partitioned as

1 = 1 2 = 1 + 1 3 = 1 + 1 + 1 4 = 1 + 1 + 1 + 1

= 2 = 1 + 2 = 1 + 1 + 2

= 2 + 1 = 1 + 2 + 1

= 3 = 1 + 3

= 2 + 1 + 1

= 2 + 2

= 3 + 1

= 4

By observation, there is

one way to partition 1.

two ways to partition 2.

four ways to partition 3.

eight ways to partition 4.

In general, there are exp2(n-1) ways to partition value n.

In the above examples,

when partitioning 1, "+" is not used.

when partitioning 2, "+" is used once.

when partitioning 3, "+" is used four times.

when partitioning 4, "+" is used twelve times.

When partitioning value n (n > 4), how many times is "+" used?

You can use the following table to validate your formula. If your

formula does not match these cases, the formula is definitely wrong.

However, matching these cases does not mean your formula is correct.

You must have a systematic way to find the formula. Do not try to

find a formula to match these values.

n 1 2 3 4 5 6 7 8

f(n) 0 1 4 12 32 80 192 448