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Template:Sequence of the Day for October 28

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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
Yesterday's SOTD * Tomorrow's SOTD

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.