Greedy frac multiples of sqrt(5): a(1)=1, sum(n>0,frac(a(n)*x))=1 at x=sqrt(5).

Terms

a(0) =1a(1) =2a(2) =5a(3) =13a(4) =17a(5) =34a(6) =305a(7) =610a(8) =1597a(9) =4181a(10) =5473a(11) =10946a(12) =98209a(13) =196418a(14) =514229a(15) =1346269a(16) =1762289a(17) =3524578a(18) =31622993a(19) =63245986a(20) =165580141a(21) =433494437a(22) =567451585a(23) =1134903170a(24) =10182505537a(25) =20365011074a(26) =53316291173a(27) =139583862445