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.

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

Links

Table of n, a(n) for n=0..54.

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

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

Sequence in context: A019965 A053148 A295798 * A158185 A095413 A134871

Adjacent sequences: A072725 A072726 A072727 * A072729 A072730 A072731

Keyword

easy,frac,nice,nonn

Author

Paul D. Hanna, Jul 09 2002

Status

approved