I have a gold one, but I want the silver one to make patterns. 
 
I was told silver axes make patterns 100% of the time, while gold is more rare? I could be wrong, but my husband told me that. He can be a noob sometimes, though, haha!
Yeah from wikia it says 
"Cutting down the tree with a normal axe has a rare chance, while the silver axe almost always produces one with a special pattern. The gold axe sometimes produces these special patterns, but not as often as the silver axe." 
I found that here: 
http://animalcrossing.wikia.com/wiki/Special_Stump_Patterns