[seqfan] k-chunks sum and division by k

[Eric Angelini]

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.