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

[Neil Sloane]

Thomas,  That is interesting, especially since, as you say, one of the
three sequences
has nothing more than the original definition.

I have added the following comment to all three sequences.

Conjecture: -3 A122858(n) - A229616(n) + 4 A282031(n) = 0 for all n. -
_Thomas Baruchel_, Jun 23 2018



Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com


On Sat, Jun 23, 2018 at 10:45 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
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>