a(n) = a(n-1) + a(n-2) + a([(n+1)/2]), where a(0) = 1, a(1) = 2, a(2) = 3.

Terms

a(0) =1a(1) =2a(2) =3a(3) =8a(4) =14a(5) =30a(6) =52a(7) =96a(8) =162a(9) =288a(10) =480a(11) =820a(12) =1352a(13) =2268a(14) =3716a(15) =6146a(16) =10024a(17) =16458a(18) =26770a(19) =43708a(20) =70958a(21) =115486a(22) =187264a(23) =304102a(24) =492718a(25) =799088a(26) =1294074a(27) =2096878a(28) =3394668a(29) =5497692