--description--

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 $\{s_1, s_2, \ldots, s_n\}$. 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 $f(25) = 18\,635\,853$.

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

--hints--

generatingPolygons() should return 697003956.

assert.strictEqual(generatingPolygons(), 697003956);

--seed--

--seed-contents--

function generatingPolygons() {

  return true;
}

generatingPolygons();

--solutions--

// solution required