a(2k) = floor(F(k)/2), a(2k+1) = ceiling(F(k)/2), where F = A000045 is the Fibonacci sequence.

Terms

a(0) =0a(1) =0a(2) =0a(3) =1a(4) =0a(5) =1a(6) =1a(7) =1a(8) =1a(9) =2a(10) =2a(11) =3a(12) =4a(13) =4a(14) =6a(15) =7a(16) =10a(17) =11a(18) =17a(19) =17a(20) =27a(21) =28a(22) =44a(23) =45a(24) =72a(25) =72a(26) =116a(27) =117a(28) =188a(29) =189