[seqfan] Re: k-chunks sum and division by k

[Eric Angelini]

merci Maximilian !

Propulsé d'un aPhone



Le 24 mars 2013 à 01:04, "Maximilian Hasler" <maximilian.hasler at gmail.com> a écrit :

> Dear Eric & SeqFans,
> to avoid duplicate efforts, just to say that I confirm
> the given values & extended them & fixed some details
> in the mentioned (and other related) sequences ;
> I will submit those not yet there (k=6,8,10) as A222256 ff
> (some unexpected events preventing me from having done this earlier)
> 
> Have a nice week-end,
> 
> Maximilian
> 
> 
> On Fri, Mar 22, 2013 at 8:16 AM, Eric Angelini <Eric.Angelini at kntv.be> wrote:
>> Hello SeqFans,
>> I've tried yesterday to build a sequence S where the sum of any k successive terms of S is divisible by k. S being the first lexicographically such sequence and S never showing twice the same integer.
>> 
>> For k odd, S is trivial:
>> S=1,2,3,4,5,6,7,8,9,10,11,12,13,...
>> 
>> For k even we have a few interesting things.
>> 
>> Let's start with k=2 and a(1)=0:
>> S=0,2,4,6,8,10,12,14,16,18,20,22,...
>> 
>> Well, no revolution here. Let's try a(1)=1:
>> S=1,3,5,7,9,11,13,15,17,19,21,23,...
>> 
>> Mmmmh.
>> 
>> For k=4 and a(1)=0 we have:
>> S=0,1,2,5,4,9,6,13,8,17,10,21,22,...
>> ... which is http://oeis.org/A114752
>>    (and which has a quite complicated definition).
>> 
>> For k=4 and a(1)=1 we get:
>> S=1,2,3,6,5,10,7,14,9,18,11,22,13...
>> ... which is http://oeis.org/A043547
>>    (a nice interspersion)
>> 
>> For k=6 and a(1)=0 I get (by hand) the new seq:
>> S=0,1,2,3,4,8,6,7,14,9,10,20,...
>> 
>> Explanation:
>> The 1st chunk of 6 consecutive integers 0->8 has sum 18,
>>                               which is divisible by 6
>> The 2nd chunk of 6 consecutive integers 1->6 has sum 24,
>>                               which is divisible by 6
>> The 3rd chunk of 6 consecutive integers 2->7 has sum 30,
>>                               which is divisible by 6
>> Etc.
>> 
>> For k=6 and a(1)=1 I get (again, by hand) the new seq:
>> S=1,2,3,4,5,9,7,8,15,...
>> 
>> I guess there is a possible new family of seq if we try
>> k=8,10,12,14,16,... for values a(1)=0 and a(1)=1.
>> 
>> Interesting patterns might be found...
>> Best,
>> É.
>> 
>> 
>> _______________________________________________
>> 
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