a(n) = F(n+8) - (1/6)*(n^4-2*n^3+26*n^2+47*n+132) where F(i) = Fibonacci numbers (A000045).

Terms

a(0) =0a(1) =0a(2) =0a(3) =0a(4) =1a(5) =8a(6) =35a(7) =113a(8) =303a(9) =717a(10) =1552a(11) =3145a(12) =6062a(13) =11242a(14) =20230a(15) =35554a(16) =61335a(17) =104274a(18) =175249a(19) =291899a(20) =482805a(21) =794255a(22) =1301190a(23) =2124915a(24) =3461756a(25) =5629428a(26) =9142060a(27) =14831588a(28) =24044173a(29) =38958012