Intended for: October 28, 2012
Timetable
 First draft entered by Alonso del Arte on August 28, 2011 ✓
 Draft reviewed by Daniel Forgues on October 28, 2012 ✓
 Draft to be approved by September 28, 2012
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A026150 (n + 1) / A002605 (n):
Convergents to
.
 { , , , , , , , , ⋯}
Obviously these fractions can be expressed in lower terms. But by leaving them as is, I wish to highlight that both the numerators
and the denominators
are obtained by
recurrence relations of order
2 (actually, the recurrences are the same, only the initial conditions differ):
a (0)  = 1, a (1) = 1; 
a (n)  = 2 (a (n − 1) + a (n − 2)), n ≥ 2. 

b (0)  = 0, b (1) = 1; 
b (n)  = 2 (b (n − 1) + b (n − 2)), n ≥ 2.


(The
0th term of the sequence would be
:= ∞.)
Note also the recurrence (involving only the previous numerator/denominator)
a (1)  = 1, b (1) = 1; 
a (n)  = a (n − 1) + 3 b (n − 1), n ≥ 2. 
b (n)  = a (n − 1) + b (n − 1), n ≥ 2.

