%I #4 Mar 30 2012 18:36:32

%S 1,2,3,5,5,8,7,8,12,13,11,12,13,19,19,21,17,18,19,19,21,29,31,30,31,

%T 34,27,26,29,29,31,30,31,34,46,45,50,46,49,49,50,55,41,44,41,43,47,46,

%U 45,50,46,49,49,50,55

%N Numerator of rationals >= 1 whose continued fractions consist only of 1's and 2's, in ascending order by the sum of the continued fraction terms and descending by lowest order continued fraction terms to highest.

%F a(F(n)+F(n-3)+m) = a(F(n-1)+m) + a(F(n-3)+m) when 0<m<=F(n-2), n>2; a(F(n)+m) = 2*a(F(n-2)+m) + a(F(n-4)+m) when 0<m<=F(n-3), n>3; where a(0)=1, a(F(n)-1) = F(n) = n-th Fibonacci number; a(F(2n-1)) = n-th Pell number.

%e n: a(n)/A072729(n) has continued fraction:

%e 0: 1/1 = [1]

%e 1: 2/1 = [2]

%e 2: 3/2 = [1;2]

%e 3: 5/2 = [2;2]

%e 4: 5/3 = [1;1,2]

%e 5: 8/3 = [2;1,2]

%e 6: 7/5 = [1;2,2]

%e 7: 8/5 = [1;1,1,2]

%e 8: 12/5 = [2;2,2]

%e 9: 13/5 = [2;1,1,2]

%e 10: 11/8 = [1;2,1,2]

%e 11: 12/7 = [1;1,2,2]

%e 12: 13/8 = [1;1,1,1,2]

%e 13: 19/8 = [2;2,1,2]

%e 14: 19/7 = [2;1,2,2]

%e 15: 21/8 = [2;1,1,1,2]

%e 16: 17/12= [1;2,2,2]

%e 17: 18/13= [1;2,1,1,2]

%e 18: 19/11= [1;1,2,1,2]

%e 19: 19/12= [1;1,1,2,2]

%e 20: 21/13= [1;1,1,1,1,2]

%Y Cf. A072729, A071585, A071766, A072726, A072727, A000129.

%K easy,frac,nice,nonn

%O 0,2

%A _Paul D. Hanna_, Jul 09 2002