a(1)=1; for n>=2, a(n)=sum(1<=i<=j<=n-1, gcd(a(i),a(j))).

Terms

a(0) =1a(1) =1a(2) =3a(3) =8a(4) =19a(5) =42a(6) =92a(7) =194a(8) =398a(9) =808a(10) =1638a(11) =3334a(12) =6686a(13) =13392a(14) =26842a(15) =53708a(16) =107450a(17) =214952a(18) =430148a(19) =860340a(20) =1720826a(21) =3441688a(22) =6883446a(23) =13766950a(24) =27533998a(25) =55068040a(26) =110136196a(27) =220272460a(28) =440545040a(29) =881090532