[seqfan] Re: The mysterious Layman sequences

[Rémy Etc]

hi Neil,

> 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!
this formula is true at least up to n=5000.

best regards,

Rémy

Le jeu. 13 mai 2021 à 20:03, Neil Sloane <njasloane at gmail.com> a écrit :
>
> 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
>
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