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 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 $$\begin{array}{ccc}1&amp;2&amp;3\\4&amp;5&amp;6\\7&amp;8&amp;9 \end{array}$$ 

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.
Multiply the amount of nodes(9) by the number of rows and columns (9) and then multiply that by the amount of starts(9).

It would look like this: $9\times 9=81$, then, $81\times 9=729$, so it would be 729 possible answers that some one could use. Wait!, but you have to use 4 at a minimum so you multiply that by 4 like this:  $729\times 4=2916$

$2,916$ possible answers!