a(n) = floor((1/2 * (sqrt(2) + 1 + sqrt(2*sqrt(2) - 1)))^n ).

Terms

a(0) =1a(1) =1a(2) =3a(3) =6a(4) =12a(5) =23a(6) =44a(7) =84a(8) =158a(9) =297a(10) =561a(11) =1056a(12) =1989a(13) =3746a(14) =7056a(15) =13287a(16) =25023a(17) =47125a(18) =88746a(19) =167127a(20) =314735a(21) =592710a(22) =1116193a(23) =2102019a(24) =3958531a(25) =7454720a(26) =14038755a(27) =26437833a(28) =49787820a(29) =93760597