Generate Toothpick Sequence
CJam, 99 93 bytes

This got rather long...

"\ "_W%]{{Sf+W%z}4*2ew{2fewz{_sa"\//\\"4S**4/^_,3={:.e&gt;W%2/\}&amp;;}%z{\)a@.e&gt;+}:Ff*}%{F}*}q~(*N*


Test it here. If you want to test larger inputs, like the 89 on Wikipedia, Dennis's TryItOnline uses the much faster Java interpreter under the hood and can handle inputs like that in a few seconds.

I'm sure there is a lot of room for improvement, and I'll add an explanation once I'm happier with the score...

Here is the output for N = 13:

            \             
            /\            
           /\             
          / /\            
        \/\/\ \ \         
         \ \/\/\/\        
           /\/\/          
          / /\ \          
    \    /\/\ \     \     
    /\  /  \/\/\    /\    
   /\  /\  /\/  \ \/\     
  / /\/ /\/ /\  /\ \/\    
\/\/\/\/\/\/\ \/\ \/\ \ \ 
 \ \ \/\ \/\ \/\/\/\/\/\/\
    \/\ \/  \/ /\/ /\/ /  
     \/\ \  /\/  \/  \/   
    \/    \/\/\  /  \/    
     \     \ \/\/    \    
          \ \/ /          
          /\/\/           
        \/\/\/\ \         
         \ \ \/\/\        
            \/ /          
             \/           
            \/            
             \            


For my own reference when golfing this further, some other ideas:

"\ "_W%]{{Sf+W%z}4*2few2ew::.{+_a"\//\\"4S**4/^_,3={:.e&gt;W%\}&amp;;2/}:z{\)a@.e&gt;+}ff*{\)a@..e&gt;+}*}ri(*N*




"\ "_W%]{{Sf+W%z}4*2ew{2fewz{_sa"\//\\"4S**4/^_,3={:.e&gt;W%2/\}&amp;;}%{.{\)a@.e&gt;+}}*}%{\)a@.e&gt;+}*}q~(*N*