Counting ethers and choosing a route

You will write functions that count the isomers of a formula $\text{C}_n\text{H}_{2n+2}\text{O}$, and decide which route to an ether is better.

The idea. An ether is a pair of alkyl groups. The number of alkyl groups with 1 to 6 carbons is 1, 1, 2, 4, 8 and 17. An ether with $n$ carbons has two groups of $a$ and $b$ carbons, with $a + b = n$. Two pairs that differ only in order are the same ether. In a Williamson synthesis the halide should be the less crowded one.

Your tasks.

1. Complete count_ethers(n) so that it counts the ethers, using the table alkyl.
2. Complete count_isomers(n) so that it adds the alcohols to the ethers.
3. Complete better_route(halide_a, halide_b) so that it returns "a" or "b", whichever route uses the less hindered halide.