[seqfan] Re: The mysterious Layman sequences

[cwwuieee]

Neil, I think you mean c=0.36704=31*37/5^5.If my calculations are correct, the recurrence is satisfied for the first 4000 terms.Chaiwah
-------- Original message --------From: Neil Sloane <njasloane at gmail.com> Date: 5/13/21  2:03 PM  (GMT-05:00) To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu> Subject: [seqfan] The mysterious Layman sequences Dear Seq Fans,Rick Mabry has made some interesting comments about a family of sequencescalled "Ratio-Determined Insertion Sequences" contributed many years ago bythe late John Layman. I propose to do some major editing of them, but firstI would like to know the answer to a specific question. It needs somecomputing 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) = 0.36704exactly.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 betweenevery 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, 30Assuming that no one can prove this conjecture, I propose to replace theexisting definition with the recurrence, and state it as a conjecture thatit agrees with the sequence produced by the insertion rule. (I won't takethe space here to explain why I want to do this.)But first I would like to be sure that the conjecture is true.  So couldsomeone please generate a lot of terms using the present definition (theinsertion 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 ratherstrange constant, and why should  the resulting sequence be explained bythe quite simple recurrence a(n) = 10*a(n-2) - a(n-4) ?  So I want anumerical verification, as far out as is convenient, before I accept it!There are two text files written by Layman attached to the sequence. I havenot studied them carefully, maybe they contain the answer to the question.Best regardsNeilNeil 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.comEmail: njasloane at gmail.com--Seqfan Mailing list - http://list.seqfan.eu/