[Fwd: Re: Puzzle sequence and A109652]

[Joseph Biberstine]

Here's a message I got from Graeme McRae; I agree with every word.

- JRB

-------- Original Message --------
Subject: Re: Puzzle sequence and A109652
Date: Sat, 5 Aug 2006 20:33:41 -0700
From: Graeme McRae <g_m at mcraefamily.com>
To: Joseph Biberstine <jrbibers at indiana.edu>
References: <20060806014414.31097.qmail at web38201.mail.mud.yahoo.com>
<44D55846.5080903 at indiana.edu>

For some reason, messages I send to seqfan get bounced.  I must have the
wrong email address.  Here's what I tried to send:

By "primes aligned with...", the author of A109652 meant this:  Place the
sequence of primes side-by-side with the sequence A003849.  For each zero in
A003849, select the corresponding prime number and make it part of A109652.

Apart from being subject to misunderstanding, this definition has another
flaw.  Primes are offset 1, and A003849 is offset 0.  Correcting for that
difference in offsets, the definition might be better clarified this way:

A109652 is the sequence of numbers, n, such that n=A40(k+1) and A3849(k)=0
for k=0,1,2,3,...

This means the same as A40(A201), except the offset would then need to be
changed to 1.

As the offsets are confused as it is, changing it to 1 would not make
matters worse, and the definition A40(A201) is certainly easier to
understand, and not ambiguous, so I would suggest changing the definition,
but leaving the "primes aligned with..." as a comment, further clarified to
say what that means in a manner similar to what I suggested, above.


----- Original Message -----
From: "Joseph Biberstine" <jrbibers at indiana.edu>
To: "zak seidov" <zakseidov at yahoo.com>
Cc: <seqfan at ext.jussieu.fr>
Sent: Saturday, August 05, 2006 7:47 PM
Subject: Re: Puzzle sequence and A109652


> While toying with the puzzle seq I came across A109652, which appears
> to be inconsistent with its definition.  Depending on the offset, my
> interpretation of the definition, and Cloitre's formula, one would
> expect at least one of the following sequences to be all zero:
>
> A3849(A109652 - 2) = {0, 0, 0, 0, 1, ...}
> A3849(A109652 - 1) = {1, 1, ...}
> A3849(A109652) = {0, 0, 0, 0, 1, ...}
> A3849(A109652 + 1) = {0, 1, 0, ...}
> A3849(A109652 + 2) = {1, 0, 1, 0, ...}
>
> These were calculated using the following formula from A3849:
> "a(n) = floor((n+2)*r)-floor((n+1)*r) where r=phi/(1+2*phi) and phi is
> the Golden Ratio. - Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 10 2003"
>
> *However*, a fitting definition for A109652 seems to be A40(A201).  All
> terms agree.
>
> - JRB
>