a(n) = Sum_{k=0..floor(n/3)} Stirling2(n - 2*k,n - 3*k).

Terms

a(0) =1a(1) =1a(2) =1a(3) =1a(4) =2a(5) =4a(6) =7a(7) =12a(8) =23a(9) =47a(10) =95a(11) =192a(12) =402a(13) =869a(14) =1898a(15) =4181a(16) =9379a(17) =21431a(18) =49556a(19) =115770a(20) =273919a(21) =656476a(22) =1590061a(23) =3888783a(24) =9608337a(25) =23980678a(26) =60402964a(27) =153469477a(28) =393325442a(29) =1016628823