Number of permutations of [n] whose non-fixed points are not neighbors.

Terms

a(0) =1a(1) =1a(2) =1a(3) =2a(4) =4a(5) =9a(6) =19a(7) =45a(8) =107a(9) =278a(10) =728a(11) =2033a(12) =5749a(13) =17105a(14) =51669a(15) =162674a(16) =520524a(17) =1724329a(18) =5807143a(19) =20146861a(20) =71048431a(21) =257139686a(22) =945626800a(23) =3558489633a(24) =13599579817a(25) =53060155137a(26) =210124405097a(27) =847904374466