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