Jelly, 1515 9 bytes
-,3,3,1ṫḢạ’×12Ɗ’3cạ×12$Ʋ
A monadic link taking \$n\$ as its argument and returning \$a(n)\$.
Based on @LevelRiverSt’s clever Ruby answer so be sure to upvote that one too!
Thanks to @Grimmy for saving 6 bytes!
Explanation
-,3,3,1ṫ ’ | Sublist ofSubtract -1,3,3,1 starting at
the nth character (empty if n>4)
Ʋ | Following as a Ḣmonad
3c | Head (zero if- emptyNumber list)
of ways of picking (n-1) items from 3
ạ $ Ɗ | - Absolute difference from:
×12 | ’×12 - |Multiply (n-1) × 12