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

[jean-paul allouche]

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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