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Old 2004-10-08, 04:48   #2
Zeta-Flux
 
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May 2003

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Here are some possible lines of attack that I found. (You can plug them into LaTeX if you can't follow the notation).

Define A_m = ( (1+\sqrt{2})^m + (1-\sqrt{2})^m )/2. Then this gets rid of the binomial stuff. And if we plug in m = k_n we get exactly the same numbers as you defined earlier.

There are some interesting recurrence relations for the A_m. Look at:
Code:
A_1  =  1  
              >   0
A_2  =  1          >  2
              >   2         >  0
A_3  =  3          >  2          >  4
              >   4         >  4           >0
A_4  =  7          >  6          >  4
              >  10         >  8
A_5  = 17         > 12
              >  24
A_6  = 41
where the number immediately following > denotes the difference of the two previous numbers (to the left of >). Notice that every row is 2 times the row that occurs two places back. This pattern continues. So one can reconstruct this pattern using this fact and that the first part looks like:

1
> 0


Hope that gives you something new to ponder. (But I don't know if it will solve your problem.) Where did you get those k_n numbers from?

Best,
Zeta-Flux

Last fiddled with by Zeta-Flux on 2004-10-08 at 04:51
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