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# Template:Sequence of the Day for October 28

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

The line below marks the end of the <noinclude> ... </noinclude> section.

A026150 (n + 1) / A002605 (n): Convergents to
 2√  3
.
{
 1 1
,
 4 2
,
 10 6
,
 28 16
,
 76 44
,
 208 120
,
 568 328
,
 1552 2448
, }
Obviously these fractions can be expressed in lower terms. But by leaving them as is, I wish to highlight that both the numerators
 a (n)
and the denominators
 b (n)
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
 1 0
:= ∞
.)

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.