a(n) = ceiling( 1/2 + (Sum_{i=0..n-1}C(n,i)*C(n,i+1))/2^(n+1) ).

Terms

a(0) =1a(1) =2a(2) =3a(3) =4a(4) =7a(5) =13a(6) =23a(7) =44a(8) =83a(9) =159a(10) =306a(11) =590a(12) =1144a(13) =2220a(14) =4317a(15) =8408a(16) =16399a(17) =32023a(18) =62601a(19) =122498a(20) =239924a(21) =470304a(22) =922612a(23) =1811217a(24) =3558035a(25) =6993883a(26) =13755529a(27) =27068914a(28) =53294747a(29) =104979547