## Generating polygons

### Problem 382

Published on Saturday, 28th April 2012, 08:00 pm; Solved by 245
A **polygon** is a flat shape consisting of straight line segments that are joined to form a closed chain or circuit. A polygon consists of at least three sides and does not self-intersect.

A set S of positive numbers is said to *generate a polygon* P if:

- no two sides of P are the same length,
- the length of every side of P is in S, and
- S contains no other value.

For example:

The set {3, 4, 5} generates a polygon with sides 3, 4, and 5 (a triangle).

The set {6, 9, 11, 24} generates a polygon with sides 6, 9, 11, and 24 (a quadrilateral).

The sets {1, 2, 3} and {2, 3, 4, 9} do not generate any polygon at all.

Consider the sequence s, defined as follows:

- s
_{1}= 1, s_{2}= 2, s_{3}= 3 - s
_{n}= s_{n-1}+ s_{n-3}for`n`> 3.

Let U_{n} be the set {s_{1}, s_{2}, ..., s_{n}}. For example, U_{10} = {1, 2, 3, 4, 6, 9, 13, 19, 28, 41}.

Let f(`n`) be the number of subsets of U_{n} which generate at least one polygon.

For example, f(5) = 7, f(10) = 501 and f(25) = 18635853.

Find the last 9 digits of f(10^{18}).