[seqfan] Possible other integer sequences for Šindel sequences?

[C.R. Drost]

Hi there!

I recently became interested in the Orloj sequence

https://oeis.org/A028355

which is really a sequence of decimal interpolations of sequences. There is
a nice paper linked about all of the other sequences than 1-2-3-4-3-2 that
have this remarkable property, like 1-2-2, where successive takes of the
repetition of the sequence can sum to the counting numbers,

1, 2, 2+1, 2+2, 1+2+2, 1+2+2+1, 2+2+1+2, 2+1+2+2+1, ...

the key to the proof being that every later number like 11 can be viewed as
(the construction for 1) + (2 periods of the sequence). The paper calls
these Šindel sequences and has a nice set of constructions for them that do
not require enumerating something like n! sequences of length n and
checking for a proof that they cover all the integers, but just starts from
the sum: it turns out that the sum of the digits uniquely specifies the
sequence up to a process of "breakup" where e.g. 1-2-2 becomes 1-2-1-1 or
1-1-1-2 and eventually 1-1-1-1-1; you can always break apart one term in
any Šindel-sequence into two more that sum to the same.

It looks like these are pleasantly unrelated to anything else in OEIS and
I'd like to add some integer sequences, I am just struggling with how to
define what I am adding :)

I could do the number of primitive Šindels of length n, this seems to stay
low and erratic,

0, 1, 2, 1, 4, 0, 3, 1, 5, 1, 1, 2, 7, 0, 2, 1, 7, 0, 3, 1, 2, 2, 5, 0, 12,
0, 0, 2, 4, 0, 4, 2, 8, 1, 1, 0, 8, 1, 2, 0, 5, 0, ...

One thing I don't like about this sequence is that the logical outpouring
of the math for primitive Šindels causes the primitives to include a bunch
of things that I think are kind of "cheating", for example those two
2-periodic sequences are 1-1-1-1-1-1-1-... and 1-2-1-2-1-2-... and I feel
like the first has already been counted in some sense; it is not *strictly*
2-periodic. (That is, I consider the sequences to be infinite sequences,
not finite chunks to be repeated.) So if I am chasing strict periodicity
then I need to abandon the notion of primitive Šindels and instead get a
count of strictly-n-periodic Šindels,

0, 1, 1, 1, 4, 10, 14, 29, 62, 146, 293, 610, 1139, 2070, ...

would either of these be interesting submissions to OEIS? Should I abandon
the notion of strict periodicity so that each term contains the sum of the
term for that number in the above sequence plus the terms for all of its
factors? Etc.

-- Chris