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, "3" is not used.

when partitioning 2, "3" is not used.

when partitioning 3, "3" is used once.

when partitioning 4, "3" is used twice.

When partitioning value n (n > 4), how many times is "3" used?

The answer is not exp2(n-2). 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 4 5 6 7 8 9

f(n) 2 5 12 28 64 144