a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+3, k).

Terms

a(0) =1a(1) =1a(2) =5a(3) =6a(4) =17a(5) =23a(6) =50a(7) =73a(8) =138a(9) =211a(10) =370a(11) =581a(12) =979a(13) =1560a(14) =2575a(15) =4135a(16) =6755a(17) =10890a(18) =17700a(19) =28590a(20) =46356a(21) =74946a(22) =121380a(23) =196326a(24) =317797a(25) =514123a(26) =832025a(27) =1346148a(28) =2178293a(29) =3524441