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

Sun Feb 28 20:26:05 CET 2021

If mathematical notation is to be of any use at all,
a(n) = c*n^2  is not the same as a(n) ~ c*n^2

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 2:18 PM jean-paul allouche <
jean-paul.allouche at imj-prg.fr> wrote:

> But if I am not mistaken, if f(n) tends to infinity, then [f(n)] equiv f(n)
> since the difference is bounded..
> Now n^2/4 + n equiv n^2/4 and n^2/4 + n equiv n^2/4.
> So if we understand the "=" sign to be an "equiv" sign, we do have
> a(n) equiv n^2/4.
>
> best
> jp
>
> Le 28/02/2021 à 20:11, Neil Sloane a écrit :
> >> 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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