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

Terms

a(0) =-1a(1) =2a(2) =1a(3) =7a(4) =14a(5) =41a(6) =103a(7) =274a(8) =713a(9) =1871a(10) =4894a(11) =12817a(12) =33551a(13) =87842a(14) =229969a(15) =602071a(16) =1576238a(17) =4126649a(18) =10803703a(19) =28284466a(20) =74049689a(21) =193864607a(22) =507544126a(23) =1328767777a(24) =3478759199a(25) =9107509826a(26) =23843770273a(27) =62423800999a(28) =163427632718a(29) =427859097161