Eccoci.

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 $n \gt 3$.

Let $U_n$ be the set $\{s_1, s_2, \dots, s_n\}$. 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 $f(10^{18})$.