a(n) = 0^n + 1 - F(n-1)^2 - F(n)^2, where F = A000045.

Terms

a(0) =1a(1) =0a(2) =-1a(3) =-4a(4) =-12a(5) =-33a(6) =-88a(7) =-232a(8) =-609a(9) =-1596a(10) =-4180a(11) =-10945a(12) =-28656a(13) =-75024a(14) =-196417a(15) =-514228a(16) =-1346268a(17) =-3524577a(18) =-9227464a(19) =-24157816a(20) =-63245985a(21) =-165580140a(22) =-433494436a(23) =-1134903169a(24) =-2971215072a(25) =-7778742048a(26) =-20365011073a(27) =-53316291172a(28) =-139583862444