a(n) = 2*a(n-1) + a(n-2) - [a(n-2)/2] - [a(n-4)/2] - [a(n-5)/2] where [k] := floor(k).

Terms

a(0) =1a(1) =1a(2) =3a(3) =7a(4) =16a(5) =36a(6) =79a(7) =172a(8) =373a(9) =806a(10) =1742a(11) =3762a(12) =8123a(13) =17538a(14) =37864a(15) =81745a(16) =176480a(17) =381003a(18) =822545a(19) =1775788a(20) =3833737a(21) =8276627a(22) =17868350a(23) =38575848a(24) =83281109a(25) =179794961a(26) =388157989a(27) =837991360a(28) =1809133237a(29) =3905724120