[seqfan] Re: Composite Hurwitz Zeta formulas for A007494 & A032766

[Fred Lunnon]

  Perhaps this might be a consequence of Hasse's series, combined with
that notorious Euler sum --- see
    https://en.wikipedia.org/wiki/Hurwitz_zeta_function
    https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF
If so, it should generalise to binomial coefficients.

WFL



On 11/15/17, Andras via SeqFan <seqfan at list.seqfan.eu> wrote:
> I've been trying to break down & interrogate the function for the past
> couple of days to obtain a better understanding of it and found something
> unrelated to the sequence itself involving the Hurwitz Zeta function so I am
> putting it here for lack of anywhere else better.
> a(n) = -(1/12)  +  -HurwitzZeta(-1, n)     yields the nth Triangular number
> as a whole number.(https://i.imgur.com/Ld3e7yC.png for the domain colored
> plot)
>
> I figured I'd ask before editing the page: Is this following Mathematica
> snippet worth including on A000217?
> Table[-(1/12)  +  -HurwitzZeta[-1, Z], {Z, 50}]
>
>    On Saturday, November 11, 2017, 5:26:49 PM EST, Andras <andras at yahoo.com>
> wrote:
>
>  A293687 and A294921 have been submitted along with their domain colored
> plots. Feel free to make any modifications you feel are necessary since this
> is my first time submitting sequences on my own.
>  On Friday, November 10, 2017, 3:22:03 AM EST, Neil Sloane
> <njasloane at gmail.com> wrote:
>
>  Yes, go ahead and submit the two "squared" versions, and if possible upload
> the nice pictures(or, at least, give links to the pictures - but it would be
> better for the long term if they were on the OEIS server)
> Best regardsNeil
> 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.comEmail: njasloane at gmail.com
>
> On Thu, Nov 9, 2017 at 8:58 PM, Andras via SeqFan <seqfan at list.seqfan.eu>
> wrote:
>
> I have two composite Hurwitz Zeta function formulas for "Numbers that are
> congruent to 0 or 2 mod 3." and "Numbers that are congruent to 0 or 1 mod
> 3." that I thought might be of some interest.
>
> My question is if these 2 sequences merit their own entries (which would be
> A007494 & A032766 squared & omitting the Sqrt function in the formula) & if
> so what they should be named, or alternatively if they should instead be
> included in the formula sections for A007494 & A032766. Here are some domain
> colored visualizations of the functions: https://i.imgur. com/GGPG2E0.jpg ,
> also, if anyone knows what these types of composite zeta functions are
> named, that'd be greatly appreciated!
> Mathematica Code:
> ζ = HurwitzZeta
>
> A007494 =  Function[{Z},   Sqrt[4 - 9/2 ζ[-1, 2/3 + Z] - 3 ζ[-1, 5/6] +
> 3 ζ[-1, 4/3] + 9/2 ζ[-1, 5/3] +     3 (-1)^Z ζ[-1, 5/6 + Z/2] -     3 (-1)^Z
> ζ[-1, 1/6 (2 + 3 Z)]]]
> A032766 =  Function[{Z},   Sqrt[9 - 9/2 ζ[-1, 4/3 + Z] - 3 ζ[-1, 7/6] +
> 3 ζ[-1, 5/3] + 9/2 ζ[-1, 7/3] +     3 (-1)^Z ζ[-1, 7/6 + Z/2] -     3 (-1)^Z
> ζ[-1, 1/6 (4 + 3 Z)]]]
> Table[N[A007494[n]], {n, 0, 25}]
> Table[N[A032766[n]], {n, 0, 25}]
>
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