a(n) = Sum_{k=0..n} C(floor((n+1)/2),floor((k+1)/2))*2^k.

Terms

a(0) =1a(1) =3a(2) =7a(3) =21a(4) =37a(5) =123a(6) =187a(7) =681a(8) =937a(9) =3663a(10) =4687a(11) =19341a(12) =23437a(13) =100803a(14) =117187a(15) =520401a(16) =585937a(17) =2667543a(18) =2929687a(19) =13599861a(20) =14648437a(21) =69047883a(22) =73242187a(23) =349433721a(24) =366210937a(25) =1763945823a(26) =1831054687a(27) =8886837981a(28) =9155273437a(29) =44702625363