How many ways are there to go from $A$ to $B$:
Without passing any point twice?
Without passing any point twice or going to left side?
I need an answer without using sth like $a_n=a_{n-1}+2a_{n-2}$. (I don't even know what do they call it :)) Using numbers I got the answer $88$ for the second one. I think one of the answers is $384$ but I am not sure which one.
It is easy to compute an answer for the second question by hand for each point moving from left to right, as shown at picture below, just sum for each point up to 3 numbers on point to the left.
In general case of $n^2$ small triangles we can get recurrent relation and understand that it is the same as oeis.org/A006318, that has no closed form (i. e., without recurrent relations or summation). But OEIS confirms answer 90 :)
The first question is not so easy. I have computed 5 first values (using program) and didn't find this sequence in OEIS, so it doesn't seem to be well-known. The answer for your question is 2726.