a(n) is the number of permutations that avoid 312 and 4321 and whose square avoids 321.

Terms

a(0) =1a(1) =1a(2) =2a(3) =5a(4) =11a(5) =23a(6) =50a(7) =109a(8) =236a(9) =511a(10) =1108a(11) =2402a(12) =5206a(13) =11284a(14) =24459a(15) =53016a(16) =114914a(17) =249081a(18) =539894a(19) =1170243a(20) =2536551a(21) =5498082a(22) =11917326a(23) =25831309a(24) =55990457a(25) =121361689a(26) =263056605a(27) =570186341a(28) =1235903062a(29) =2678872272