One$-$dimensional solitaire is played on an infinite strip of 'holes' in which there are

initially some pegs. A move is a hop of one peg over another $($adjacent$)$ peg into a hole; the peg that has been hopped over is removed. You win the game if you can remove all pegs but one. Some games are winnable and some are not.

For instance, with $7$ pegs $($pegs denoted by $*,$ and holes denoted by $)$

$000***0****000$ can be won but $000***000****000$ cannot.

Task:

By hand work out the number of distinct winnable starting positions with $7$ pegs. By

'distinct' we mean that we regard a position and its mirror image as being the same,

e$.$g$.****$ is the same as $****.$

Then:

Write a program to find the number of distinct winnable starting positions with

$20$ pegs.

Print out the winnable starting positions in an easily understandable way.