a(n) = floor((2*n - 3 + sqrt(1 + 4*n!))/2).

Terms

a(0) =0a(1) =2a(2) =4a(3) =7a(4) =14a(5) =31a(6) =76a(7) =207a(8) =609a(9) =1913a(10) =6327a(11) =21896a(12) =78922a(13) =295272a(14) =1143549a(15) =4574158a(16) =18859692a(17) =80014850a(18) =348776594a(19) =1559776287a(20) =7147792837a(21) =33526120102a(22) =160785623566a(23) =787685471345