Enigmatic Travel

Suhan and Laina live in a city that has $L$ locations. Every pair of locations lies the same distance apart, and each pair is joined by exactly one two-way road — so the road map is a complete graph on $L$ vertices. Driving along any single road costs exactly $1$ universal joule.

They love wandering the city together and keep every trip secret: no one knows where they start, where they finish, or which roads they take. A trip may begin at any location and end at any location — the end may be the same as the start — and it may follow any sequence of roads they like. (For instance, when the trip is a simple cycle, the start and the end coincide.)

Given the number of locations $L$, compute the expected (average) cost of a single trip under three different assumptions about what kind of trip it is. You may assume the cost of a trip never exceeds $L$.

The input consists of several lines. Each line holds a single integer $L$ ($2 < L \le 15$), the number of locations. The input ends with a line containing $0$, which must not be processed.

For every input line, print one line with three floating-point numbers $F_1$, $F_2$, $F_3$, each rounded to exactly four digits after the decimal point:

• $F_1$ is the expected cost of an arbitrary trip (any sequence of roads, i.e. a walk).
• $F_2$ is the expected cost of a trip guaranteed to be a simple path (no location is visited twice).
• $F_3$ is the expected cost of a trip guaranteed to be a simple cycle (it returns to the start and repeats no other location).

Every cost is measured in universal joules.