From Bob L. Peterson's post to geometry.puzzles:

---------------------
From the star at the top how many paths are there without doubling
back to the dollar sign. If you find the answer in the first few days
just please post Merry Christmas. I should have used some other symbol
but for most it is very hard to ignore at Christmas but lets try
together. 
                                 *
                                / \
                               #   # 
                              / \ / \
                             #   #   #
                            / \ / \ / \
                           #   #   #   #
                          / \ / \ / \ / \
                         #   #   #   #   # 
                        / \ / \ / \ / \ / \
                       #   #   #   $   #   #
                                 ||       
                                 ||         

Bob L. Petersen
---------------------

I believe the answer will be:

Routes(4, 2)
where:
Routes(n, m) = Routes(n-1, m) + Routes(n, m-1)
and
Routes(n <= 0, m) = 1
and
Routes(n, m <= 0) = 1.


Now i can't do actual math but i can write javascript, so here goes-



m: 
 
n: