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A072728
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.
3
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
OFFSET
0,2
FORMULA
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.
EXAMPLE
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]
CROSSREFS
KEYWORD
easy,frac,nice,nonn
AUTHOR
Paul D. Hanna, Jul 09 2002
STATUS
approved