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.
|
|