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

Terms

a(0) =1a(1) =1a(2) =1a(3) =2a(4) =2a(5) =2a(6) =3a(7) =3a(8) =4a(9) =5a(10) =6a(11) =7a(12) =8a(13) =10a(14) =12a(15) =14a(16) =17a(17) =21a(18) =25a(19) =31a(20) =37a(21) =44a(22) =54a(23) =65a(24) =78a(25) =94a(26) =114a(27) =138a(28) =166a(29) =200