Intriguing property of sequence A111698

[Diana Mecum]

Sequence Fan Group,

I discovered an intriguing fact about sequence A111698 submitted by Leroy
Quet in the process of adding sequence terms.

If there are four or more non-listed integers less than a(n) for some
arbitrary n, then we choose the smallest one which is a composite distance
from a(n), (distance here defined in steps of non-listed integers). This
smallest integer is chosen for a(n+1).

However, if there are three or fewer non-listed integers less than a(n),
then the fourth non-listed integer greater than a(n) is chosen for a(n+1).

I have found two patterns for this sequence. The first is that there is a
pattern 0,3,6,0,3,6,0,3,6,... which states the lengths of the "LessThanList"
for each term. In other words, a(6) = 12. There are six integers less than
12 which are not already listed in the sequence at this point,
{3,4,6,8,10,11}. a(7) = 3. There are no integers not already on the list
which are less than 3 at this point. a(8) = 10. There are three integers
less than 10 which are not already on the list at this point, {4,6,8}.

I have worked this out to 1200 terms and see this pattern consistently.

Also, after the 14th term, the sequence becomes regular in the following
way. The difference between successive terms is as follows:
5,-13,11,5,-13,11,...

This sequence starts out looking random but becomes regular.

Please see attached files.

Diana M.

-- 
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