Combination of smartphones' pattern password Have you ever seen this interface?   

Nowadays, it is used for locking smartphones.
If you haven't, here is a short video on it.

The rules for creating a pattern is as follows.


*

*We must use four nodes or more to make a pattern at least.  

*Once a node is visited, then the node can't be visited anymore.

*You can start at any node.

*A pattern has to be connected.

*Cycle is not allowed.


How many distinct patterns are possible?

I don't have the answer as "how to mathematically demonstrate the number of combinations". Still, if that helps, I brute-forced it, and here are the results.


*

*$1$ dot:  $9$

*$2$ dots: $56$

*$3$ dots: $320$

*$4$ dots: $1624$

*$5$ dots: $7152$

*$6$ dots: $26016$

*$7$ dots: $72912$

*$8$ dots: $140704$

*$9$ dots: $140704$


Total for $4$ to $9$ digits $:389,112$ combinations
