a(n) = Sum_{k=0..n} (-1)^k * binomial(n+2*k+2,n-k) * Fibonacci(k+1).

Terms

a(0) =1a(1) =2a(2) =2a(3) =4a(4) =18a(5) =58a(6) =143a(7) =337a(8) =868a(9) =2356a(10) =6289a(11) =16367a(12) =42435a(13) =110938a(14) =291366a(15) =764147a(16) =1999478a(17) =5230106a(18) =13690840a(19) =35853360a(20) =93880186a(21) =245769487a(22) =643381907a(23) =1684376234a(24) =4409868188a(25) =11545349723a(26) =30225984563a(27) =79132089737a(28) =207170284648a(29) =542380110476