a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026769.

Terms

a(0) =1a(1) =1a(2) =2a(3) =3a(4) =6a(5) =10a(6) =17a(7) =32a(8) =56a(9) =97a(10) =181a(11) =322a(12) =567a(13) =1053a(14) =1892a(15) =3369a(16) =6241a(17) =11286a(18) =20255a(19) =37463a(20) =68044a(21) =122809a(22) =226896a(23) =413376a(24) =749159a(25) =1382990a(26) =2525162a(27) =4590351a(28) =8468738a(29) =15487526