a(n) = Sum_{k=0..n} (-1)^k * binomial(n+k+3,n-k) * Fibonacci(k+1).

Terms

a(0) =1a(1) =3a(2) =6a(3) =12a(4) =26a(5) =57a(6) =123a(7) =264a(8) =568a(9) =1224a(10) =2637a(11) =5679a(12) =12230a(13) =26340a(14) =56730a(15) =122181a(16) =263143a(17) =566736a(18) =1220592a(19) =2628816a(20) =5661737a(21) =12193803a(22) =26262054a(23) =56561148a(24) =121816954a(25) =262359777a(26) =565049859a(27) =1216959960a(28) =2620992680a(29) =5644887960