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

Sun Feb 28 21:35:34 CET 2021

Of course you are right dear Neil.

I was thinking of a misprint in the formula
given by David because of the formulation
"so the asymptotic behaviour would be
a(n) = c*n^2 for some c right?"

best wishes
jean-paul

Le 28/02/2021 à 20:26, Neil Sloane a écrit :
> 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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