[seqfan] Re: 1^5+2^5+...+n^5 is a square

[Harvey P. Dale]

	I added a linear recurrence program in Mathematica to this sequence.
	Best,
	Harvey
 

-----Original Message-----
From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Max Alekseyev
Sent: Monday, October 22, 2012 12:18 PM
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: 1^5+2^5+...+n^5 is a square

This reduces to generalized Pell equation:
6q^2 = (2n+1)^2 - 3
where q = 2m/(n*(n+1)).
So there is no surprise that solutions satisfy linear recurrent equation.
Max

On Mon, Oct 22, 2012 at 10:23 AM, Charles Greathouse <charles.greathouse at case.edu> wrote:
> It was recently asked (on MathOverflow, I think) whether the formula on A031138:
>
> a(n) =11*(a(n-1)-a(n-2)) + a(n-3)
>
> was proved or merely conjectural. Of course it should be proved to be 
> included as it is, but would someone verify this?
>
> This is of course a 6th-degree Diophantine equation:
>
> 12m^2 = n^2 (n+1)^2 (2n^2 + 2n - 1)
>
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
>
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>
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