2 or 3 possibly new sequences from 1998

[N. J. A. Sloane]

someone would like to look into them?
 
 
Return-Path: <r.rosenthal at web.de>
X-Ids: 165
Message-ID: <479A7E51.9020504 at web.de>
Date: Sat, 26 Jan 2008 01:26:57 +0100
From: Rainer Rosenthal <r.rosenthal at web.de>
User-Agent: Thunderbird 2.0.0.9 (Windows/20071031)
MIME-Version: 1.0
To: "David W. Cantrell" <DWCantrell at sigmaxi.net>
CC: seqfan <seqfan at ext.jussieu.fr>
Subject: Investigating constant C=0.1688... in A081881 comment
References: <8CA26BDC949A167-9C8-28A5 at WEBMAIL-MC05.sysops.aol.com> <001401c858b6$4d10fbf0$6401a8c0 at yourxhtr8hvc4p> <8CA2B697B3073DF-14E0-23B7 at FWM-M29.sysops.aol.com> <4797CF15.7050802 at web.de> <d3dac270801231652i75dedbb0v25281f806197ef44 at mail.gmail.com> <011801c85e41$41c0d1e0$51913e44 at Dell> <479914FE.9040509 at web.de> <014c01c85f0a$795a93b0$51913e44 at Dell>
In-Reply-To: <014c01c85f0a$795a93b0$51913e44 at Dell>
Content-Type: text/plain; charset=ISO-8859-15
Content-Transfer-Encoding: 7bit
Sender: r.rosenthal at web.de
X-Sender: r.rosenthal at web.de
X-Provags-ID: V01U2FsdGVkX1/EOuyT+/HJI0do3sY9DdY44YvGUujigStGOZH+
	LxRONLmo/u7xNCcGNKqZDdtf+yU/l63YtCLLozdYeZaf74Im9T
	sU50ryVtQ=
X-Greylist: IP, sender and recipient auto-whitelisted, not delayed by milter-greylist-3.0 (shiva.jussieu.fr [134.157.0.165]); Sat, 26 Jan 2008 01:24:07 +0100 (CET)
X-Virus-Scanned: ClamAV 0.92/5553/Fri Jan 25 22:14:29 2008 on shiva.jussieu.fr
X-Virus-Status: Clean
X-Miltered: at shiva.jussieu.fr with ID 479A7DA7.001 by Joe's j-chkmail (http://j-chkmail.ensmp.fr)!

David W. Cantrell wrote:

>>>
>>>             a(n) = 1 + floor(C*exp(n))                 (1)
>>>
>>>    where
>>>
>>>      C = 0.1688563566671442037316797755009010341...    (2)

> In <http://www.research.att.com/~njas/sequences/A081881>,
> Benoit Cloitre had already said that "a(n) is asymptotic to C*exp(n)
> where C=0.1688...", which I suppose that has been proven. My simple
> numerical investigation seemed to indicate that
>
>               0 < a(n) - C*exp(n) < 1.                   (3)
>
> I don't know if that's always true, but if it is, then of course (1)
> holds for all n.

Based on your formula for A136617 we can do more such simple numerical
investigations in a reasonable amount of time. I did so and I found
that (3) doesn't hold for n = 165. I will demonstrate this now.

The value  D(n) = (a(n)-1)/exp(n)  has to be less than C for (**) to hold:

                   D(n)  <  C    is equivalent to (3)

For n=165 we get D(165) =
0.168856356667144203731679775500901034101503956897649222377225522714175330354...

This can't be distinguished from (2) but assuming the truth of your
A136617-formula we can compute better and better approximations for C, using
the monotonically decreasing sequence C(k) = a(k)/exp(k) whose limit is C.

Approximating C by C(k) with k > 168 leads to violations of (3), since C(169) =
0.168856356667144203731679775500901034101503956897649222377225522714175330329...
                                                                           ^^
which is slightly smaller than D(165). This shows

                  a(165) - C*exp(165) > 1

contradicting (3).

Cheers,
Rainer