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?

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.
