[seqfan] Re: A269254

[Brad Klee]

After:
>http://list.seqfan.eu/pipermail/seqfan/2017-October/018033.html
>http://list.seqfan.eu/pipermail/seqfan/2017-October/018037.html
>http://list.seqfan.eu/pipermail/seqfan/2017-October/018035.html

Actually we need the inverse of the asserted algorithm "gfun_rec*rec". A
brute force search will list divisors of each term in a particular
iteration, and search every increasing combination of divisors for a linear
recurrence such as u_k = c_{k-1}u_{k-1}+c_{k-2}u_{k-2}. When found, the
algorithm returns coefficients ( c_{k-1} , c_{k-2} ) and initial
conditions. For example:

n, signature, Initial Condition
===================
408, (408,-1) , (1,559)
702, (702,-1) , (1,791)
1298, (1298,-1) , (1,1429)
2158, (2158,-1) , (1,2339)

Remarkably, these are the only cases from Hans Havermann's list to return
positive results, and also the only cases to satisfy Andrew Hone's
criteria.

Don Reble's latest discovery makes it look entirely possible that case 123
is also a sequence of composite numbers. Comment's therein imply that the
factor sequences are linear recurrences with the following signatures:

{0, 0, 0, 0, 123, 0, 0, 0, 0, -1}

{0, 0, 0, 0, 228841255, 0, 0, 0, 0, -3461681664385, 0, 0, 0, 0,
3461681664385, 0, 0, 0, 0, -228841255, 0, 0, 0, 0, 1}

These suggest a pentasection ( ha ! ) where subsequences have factors of
order two and order four.

* * * The challenge is: Can anyone modify the tactic of

http://list.seqfan.eu/pipermail/seqfan/2017-October/018030.html

to fit this particular case. My intuition is that the induction step could
again differ in details, but that it would be possible to produce some
particular zero-sum of the UV coefficient vectors. Seeing how the induction
plays out over a few more examples could improve understanding and
confidence that the technique is actually correct.

--Brad