Number of toothpicks after n stages of 3-D toothpick structure defined in Comments.

Terms

a(0) =0a(1) =1a(2) =5a(3) =21a(4) =37a(5) =53a(6) =117a(7) =197a(8) =261a(9) =405a(10) =565a(11) =789a(12) =965a(13) =1221a(14) =1541a(15) =1941a(16) =2453a(17) =2933a(18) =3621a(19) =4389a(20) =5093a(21) =5909a(22) =6805a(23) =7925a(24) =9093a(25) =10629a(26) =12197a(27) =14133a(28) =15733a(29) =17717