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

Yesterday's SOTD * Tomorrow's SOTD

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

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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.