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Old 2004-10-08, 15:53   #3
T.Rex
 
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Feb 2004
France

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Thanks.
But I'm not sure it helps.
In fact, your operation seems to periodically (period = 2) build the same 2 series, but shifted to the bottom by 1 line. Add some lines to your table, and you'll see 2 series:
U_n : 0 1 2 5 12 29 ...
V_n : 2 2 6 14 34 82 ...
X_n = 2X_{n-1}+X_{n-2}
which are the Pell sequences.
And on the right, you'll see: 1 0 2 0 4 0 8 0 ... 0 2^i ...
So we have: V_n - V_{n-1} = 4 U_{n-1} or 4 U_n = V_n + V_{n-1} . I don't know yet if it helps.
Using the binomial stuff is one solution for studying the problem. Using the relationship between Pell numbers is another one. Don't know which will provide the proof ...
About the k_n, I'm writing a paper that will explain everything, soon.
In fact: F_n is prime <==> F_n | A_k_n , I think.
Tony
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