Big trouble with A049602 (replacing a formula)

[Thomas Baruchel]

On Sun, 29 Aug 2004 15:15:28 -0400, Paul D. Hanna <pauldhanna at juno.com> 
wrote:
> I see now that my offset for lucas(n) is wrong - the proper
> definition is:

<not very proud of myself>
I spent one afternoon trying to understand how the formula for A049602 
could
match another sequence (copy/paste...)
</not very proud of myself>

> Then your formula is correct except the funny characters at the end.
> The corrected form of your formula is:
>
> (*)
> a(n)=floor((lucas(2*n)-lucas(n))*sqrt(5)/10+(n%2)*fibonacci(n)+(1-n%2))
>
> where n%2 = n (mod 2).

you are right, this is clearer. The funny characters are a 
computer-notation
(in C and other languages: (expr)?a:b  means
   "if (expr != 0) then a else b"
but I agree with you on the fact it is not very elegant (though quite 
short).

Then, I think a best formula should be:

   a(n) = integer part of 1 + (phi^4-1) ( phi^(2n-2) - (-1)^n phi^(n-2) )
(with no offset confusion possible, and no computer-based notation ;-)
Are you agree this formula is correct (empirically found, but looking
a proof as soon as I have the time) ?
I really think it should replace my precedend formula.
(N.J.A. Sloane, if you read this message, could you do that ?)

Other point: I directly re-typed Paul Barry's formula on gp (just add a 
'*' at
the right location), and I can't figure out how it is related to the given 
sequence.
Is the formula right ?

Cordially,

-- 
Thomas Baruchel