[seqfan] Re: Need help concerning a conjecture related to A122858 A229616 A282031

[Alex Meiburg]

As a sort of starting point, how easy is it to verify that -3 A122858(n) -
A229616(n)  is always divisible by 4, and that - A229616(n) + 4 A282031(n)
is always divisible by 3?

-- Alexander Meiburg

On Sat, Jun 23, 2018 at 7:45 AM, Thomas Baruchel <baruchel at gmx.com> wrote:

> Hi,
>
> with the help of my program, I could detect the following relation:
>
> -3 A122858(n) - A229616(n) + 4 A282031(n) = 0
>
> the exact log being:
>
> A122858 A229616 A282031    -->    -3 -1 4    (26)
>> A122858 Expansion of E(k) * K(k) * (2/Pi)^2 in powers of q^2 where E(),
>> K() are complete elliptic integrals and the nome q = exp( -Pi * K(k') /
>> K(k)).
>> A229616 Expansion of (phi(-q)^3 / phi(-q^3))^2 in powers of q where phi()
>> is a Ramanujan theta function.
>> A282031 Coefficients in q-expansion of (9*E_2(q^3)-E_2(q))/8.
>>
>
> It may be interesting because the last sequence has no formula in its
> description.
> However, I am not sure being able to get the full of this conjecture.
>
> Some formulass like
>
>   a(n)=if(n<1,n==0,sumdiv(n,d, -6*d* (-1)^d - 3*
> d*[0,1,-3,4,-3,1][d%6+1]));
>   (PARI syntax)
>
> are easy do build, but someone else could certainly get much more from the
> above
> identity and probably add several lines in the description of A282031.
>
> Best regards,
>
> --
> Thomas Baruchel
>
> --
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>