a(n) = (1/4)*(3 + Sum_{k=0..n} C(4k,k)).

Terms

a(0) =1a(1) =2a(2) =9a(3) =64a(4) =519a(5) =4395a(6) =38044a(7) =334054a(8) =2963629a(9) =26499449a(10) =238414581a(11) =2155749364a(12) =19572882981a(13) =178326272881