a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026747.

Terms

a(0) =1a(1) =1a(2) =2a(3) =4a(4) =6a(5) =11a(6) =20a(7) =32a(8) =58a(9) =102a(10) =169a(11) =302a(12) =527a(13) =888a(14) =1573a(15) =2741a(16) =4661a(17) =8215a(18) =14316a(19) =24481a(20) =43023a(21) =74998a(22) =128747a(23) =225867a(24) =393838a(25) =678047a(26) =1188201a(27) =2072239a(28) =3575728a(29) =6261248