a(n) = F(n+1)^2 + F(n+1)*F(n) - F(n)^2, where F = A000045.

Terms

a(0) =1a(1) =1a(2) =5a(3) =11a(4) =31a(5) =79a(6) =209a(7) =545a(8) =1429a(9) =3739a(10) =9791a(11) =25631a(12) =67105a(13) =175681a(14) =459941a(15) =1204139a(16) =3152479a(17) =8253295a(18) =21607409a(19) =56568929a(20) =148099381a(21) =387729211a(22) =1015088255a(23) =2657535551a(24) =6957518401a(25) =18215019649a(26) =47687540549a(27) =124847601995a(28) =326855265439a(29) =855718194319