_Paul D. Hanna (pauldhanna(AT)juno.com), _, Jul 09 2002

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.

easy,frac,nice,nonn,new

easy,frac,nice,nonn,new

Paul D . Hanna (pauldhanna(AT)juno.com), Jul 09 2002

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.

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

0,2

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.

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

0: 1/1 = [1]

1: 2/1 = [2]

2: 3/2 = [1;2]

3: 5/2 = [2;2]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

easy,frac,nice,nonn

Paul D Hanna (pauldhanna(AT)juno.com), Jul 09 2002

approved