a(n) = Sum_{k=0..n} (-1)^k*F(k)*F(k+2), where F=A000045 (Fibonacci numbers).

Terms

a(0) =0a(1) =-2a(2) =1a(3) =-9a(4) =15a(5) =-50a(6) =118a(7) =-324a(8) =831a(9) =-2195a(10) =5725a(11) =-15012a(12) =39276a(13) =-102854a(14) =269245a(15) =-704925a(16) =1845483a(17) =-4831574a(18) =12649186a(19) =-33116040a(20) =86698875a(21) =-226980647a(22) =594243001a(23) =-1555748424a(24) =4073002200a(25) =-10663258250a(26) =27916772473a(27) =-73087059249a(28) =191344405191