Please refer to the image an example of so-called "green-red snake" here, starting from number 10, and finally arrives 15:
Like a krait, a green-red snake has green band or red band. You can have two or more consecutive green or red bands.
Each band start with a number $a$, end with a number $c$, and has a number $b$ in the middle.
If it's a green band, then $a+b=c$; otherwise, on a red band, $a-b=c$.
Each number may only appear once on a green-red snake.
So the picture is an example of 9 bands green-red snake using up all 19 numbers from 1 to 19. Obviously, via so we can construct a green-red snake for numbers from $1$ to $2n+1$.
Question: are there other ways to consutruct a green-red snake using up all numbers from $1$ to $19$, and how many green-red snakes are there?
More generally, how many green-red snakes are there using all numbers from $1$ to $2n+1$?