Finding "sub-palindromes".
J, 24 31 40

~.(#~(1&lt;#*]-:|.)&amp;&gt;),&lt;\\.


Sample use:

   ~.(#~(1&lt;#*]-:|.)&amp;&gt;),&lt;\\. '12131331'
┌───┬───┬───┬────┬──┐
│121│131│313│1331│33│
└───┴───┴───┴────┴──┘
   ~.(#~(1&lt;#*]-:|.)&amp;&gt;),&lt;\\. '3333'
┌──┬───┬────┐
│33│333│3333│
└──┴───┴────┘


Take that, GolfScript!