a(n) = Sum_{k=0..floor(n/4)} Stirling2(n - 3*k,n - 4*k).

Terms

a(0) =1a(1) =1a(2) =1a(3) =1a(4) =1a(5) =2a(6) =4a(7) =7a(8) =11a(9) =17a(10) =29a(11) =54a(12) =102a(13) =187a(14) =337a(15) =619a(16) =1179a(17) =2298a(18) =4488a(19) =8733a(20) =17085a(21) =33931a(22) =68407a(23) =139030a(24) =283474a(25) =580477a(26) =1198195a(27) =2496661a(28) =5241757a(29) =11061986