a(n) is the number of permutations of [n] that avoid 312 and 4321 and whose square avoids 231.

Terms

a(0) =1a(1) =1a(2) =2a(3) =5a(4) =12a(5) =26a(6) =56a(7) =125a(8) =279a(9) =618a(10) =1367a(11) =3030a(12) =6720a(13) =14896a(14) =33013a(15) =73173a(16) =162198a(17) =359525a(18) =796900a(19) =1766366a(20) =3915256a(21) =8678393a(22) =19236131a(23) =42637934a(24) =94509351a(25) =209485238a(26) =464335636a(27) =1029225640a(28) =2281335673a(29) =5056707001