Q:

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?

A:

I believe the answer can be found in OEIS.  You have to add the paths of length $4$ through $9$ on a $3\times3$ grid, so $80+104+128+112+112+40=576$
I have validated the $80$, $4$ number paths.  If we number the grid $$\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9 \end{array}$$ 
The paths starting $12$ are
$1236, 1254, 1258, 1256$
and there were $8$ choices of corner/direction, so $32$ paths start at a corner.
Starting at $2$, there are
$2145,2147,2369,2365,2541,2547,2587,2589,2563,2569$ for $10$ and there are $4$ edge cells, so $40$ start at an edge.
Starting at $5$, there are $8$ paths-four choices of first direction and two choices of which way to turn
Added per user3123's comment that cycles are allowed:  unfortunately in OEIS there are a huge number of series titled "Number of n-step walks on square lattice" and "Number of walks on square lattice", and there is no specific definition to tell one from another.  For $4$ steps, it adds $32$ more paths-four squares to go around, four places to start in each square, and two directions to cycle.  So the $4$ step count goes up to $112$.  For longer paths, the increase will be larger.  But there still will not be too many.

A:

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

A:

Was bored at work and solved it total combinations are 389432 if using 3 or more
389112 is using 4 or more.

