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: Was bored at work and solved it total combinations are 389432 if using 3 or more
389112 is using 4 or more.
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: For what it is worth, I coded this problem and found 139,880 paths meeting the criteria. I don't know of any good way of validating this answer, but I can independently verify that my code gets the right answer for the number of paths with 3 points, which is 304. To see this, note that the valencies of the 9 points look like this:
$$\begin{array}{ccc}
5 & 7 & 5 \\
7 & 8 & 7 \\
5 & 7 & 5
\end{array}
$$
This is because a node at a corner can see the 5 non-corners, a node at the centre of a side can see the 7 other nodes that are not directly opposite and the node in the centre of the square can see all the 8 other nodes. Now count paths with 3 points by considering the middle point: if it has valency $n$, then it contributes $n(n-1)$ paths of length 3 ($n$ choices for the edge going in and $n - 1$ for the edge going out). So the number of paths with 3 points is:
$$4 \times 5 \times 4 + 4 \times 7 \times 6 + 8 \times 7 = 80 + 168 + 56 = 304 $$
The integer sequences http://oeis.org/A247943 and http://oeis.org/A247944 now record what I believe to be the state of the art on this question. Anyone who disagrees with the above analysis and calculations is encouraged to comment on those sequences.
A: 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!
A: Writing this as a program in python gives the following results:
path with 1 dot => 9 combinations 
path with 2 dots => 40 combinations 
path with 3 dots => 160 combinations 
path with 4 dots => 656 combinations 
path with 5 dots => 2776 combinations 
path with 6 dots => 11776 combinations 
path with 7 dots => 50488 combinations 
path with 8 dots => 217408 combinations 
path with 9 dots => 941368 combinations
