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

[jean-paul allouche]

oops I meant (n^2+1)/4 + n/2 equiv n^2/4 and n^2/4 + n equiv n^2/4
of course
jp



Le 28/02/2021 à 20:18, jean-paul allouche a écrit :
> 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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