[seqfan] The mysterious Layman sequences

[Neil Sloane]

Dear Seq Fans,
Rick Mabry has made some interesting comments about a family of sequences
called "Ratio-Determined Insertion Sequences" contributed many years ago by
the late John Layman. I propose to do some major editing of them, but first
I would like to know the answer to a specific question. It needs some
computing help, and the answer will determine how I edit
 these sequences.

So here is the question, based on the test case A085376.
This involves a certain fraction, which is c := 31*37/(2^5*5^10) = 0.36704
exactly.

Given c, we construct a triangle of numbers, as follows. The first row is
(1,1).
Given row k, we get the next row by repeating row k, except that between
every 2 adjacent terms x and y in that row, we insert their sum x+y iff y
<= c*x.

The rows converge to a sequence, which is A085376.

There is a conjecture there that this sequence satisfies a(n) = 10*a(n-2) -
a(n-4) for n >=5, with initial terms 1, 3, 11, 30

Assuming that no one can prove this conjecture, I propose to replace the
existing definition with the recurrence, and state it as a conjecture that
it agrees with the sequence produced by the insertion rule. (I won't take
the space here to explain why I want to do this.)

But first I would like to be sure that the conjecture is true.  So could
someone please generate a lot of terms using the present definition (the
insertion rule), and check that the recurrence is satisfied?

What worries me is that c = 31*37/(2^5*5^10) = 0.36704 exactly is a rather
strange constant, and why should  the resulting sequence be explained by
the quite simple recurrence a(n) = 10*a(n-2) - a(n-4) ?  So I want a
numerical verification, as far out as is convenient, before I accept it!

There are two text files written by Layman attached to the sequence. I have
not studied them carefully, maybe they contain the answer to the question.

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com