[seqfan] Re: The gerrymandering sequence A341578 needs better explanation

[Neil Sloane]

> We have a(2*n-1) = n^2 and a(k) <= a(k+1) so the asymptotic behaviour
would
be a(n) = c*n^2 for some c right?

No, that's not right. As it says in A341578:

 What is the asymptotic behavior of a(n)? - N. J. A. Sloane
<https://oeis.org/wiki/User:N._J._A._Sloane>, Feb 20 2021. Answer from Don
Reble <https://oeis.org/wiki/User:Don_Reble>, Feb 26 2020: The lower bound
is [(n^2+1)/4 + n/2]; the upper bound is [n^2/4 + n]. Each bound is reached
infinitely often. In general the best choice for d is not unique, since d
and n/d give the same answer.

Maybe you meant a(n) = O(n^2).

By the way, A341721 is a better version of the sequence.


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 Sun, Feb 28, 2021 at 1:21 PM jean-paul allouche <
jean-paul.allouche at imj-prg.fr> wrote:

> Absolutely. If a(2n-1) = n^2 and a(k) nondecreasing, then
> a(n) is equivalent to n^2/4.
> jp
>
>
> Le 27/02/2021 à 17:38, David Corneth a écrit :
> >  From A341578: What is the asymptotic behavior of a(n)? - N. J. A. Sloane
> > <https://oeis.org/wiki/User:N._J._A._Sloane>, Feb 20 2021
> >
> > We have a(2*n-1) = n^2 and a(k) <= a(k+1) so the asymptotic behaviour
> would
> > be a(n) = c*n^2 for some c right?
> >
> >
> > On Fri, Feb 26, 2021 at 7:37 PM Neil Sloane <njasloane at gmail.com> wrote:
> >
> >> Andrew W., Jack G.,  Thank you very much for the clarification. I have
> >> revised A341578 accordingly.
> >>
> >> What is the asymptotic behavior of A341578(n)?  What is the sequence of
> d
> >> values?
> >>
> >> 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 Fri, Feb 26, 2021 at 7:13 AM Andrew Weimholt <
> andrew.weimholt at gmail.com
> >> wrote:
> >>
> >>> It's not necessarily n districts with n votes each.
> >>>
> >>> For n=6, it is better to gerrymander the 36 votes into 3 districts with
> >> 12
> >>> votes each.
> >>>
> >>> In the former case, you'd need 15 votes to win: (4,4,4,3,0,0)
> >>> In the latter case, you'd only need 14: (7,7,0)
> >>>
> >>> Andrew
> >>>
> >>> On Fri, Feb 26, 2021 at 3:51 AM Jack Grahl <jack.grahl at gmail.com>
> wrote:
> >>>
> >>>> I think the confusing part is the 'grid'. This has essentially nothing
> >> to
> >>>> do with geometry.
> >>>>
> >>>> Given n districts, each with n votes, what is the least number of
> total
> >>>> votes which allows a party to win a majority of the districts?
> >>>>
> >>>> The districts are winner-takes-all, and for an even number of
> >> districts,
> >>>> it's enough to win half the districts, and tie in one further
> district.
> >>>>
> >>>> So for 5 districts of 5 votes, one party could theoretically win with
> 3
> >>>> votes in each of 3 districts, and 0 in all other districts. For 8
> >>>> districts, 5 votes in each of 4 districts, and 4 votes in a fifth
> >>> district
> >>>> is enough.
> >>>>
> >>>> On Fri, 26 Feb 2021, 10:47 Neil Sloane, <njasloane at gmail.com> wrote:
> >>>>
> >>>>> Typo, sorry. I meant to say:
> >>>>>
> >>>>> Dear Sequence Fans,  I had another look at A341578.  I accepted it
> >>>> because
> >>>>> some of the editors looked at it, and "gerrymandering" is an
> >> extremely
> >>>>> important topic.  But after looking at it more closely, I admit I
> >> don't
> >>>>> really understand the sequence.  Could someone explain the definition
> >>>> more
> >>>>> clearly?
> >>>>>
> >>>>>
> >>>>> On Fri, Feb 26, 2021 at 4:49 AM Neil Sloane <njasloane at gmail.com>
> >>> wrote:
> >>>>>> Dear Sequence Fans,  I had another look at A3415678.  I accepted it
> >>>>>> because some of the editors looked at it, and "gerrymandering" is
> >> an
> >>>>>> extremely important topic.  But after looking at it more closely, I
> >>>>> admit I
> >>>>>> don't really understand the sequence.  Could someone explain the
> >>>>> definition
> >>>>>> more clearly?
> >>>>>>
> >>>>> --
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> >>>>>
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