[seqfan] Re: A004249, A007516

[franktaw at netscape.net]

-----Original Message-----
From: Maximilian Hasler <maximilian.hasler at gmail.com>

>On Wed, Jun 10, 2009 at 3:00 PM, <franktaw at netscape.net> wrote:
>> I would say that A007516 is clearly erroneous.  Compare A014221.
>
>the sequence
>
>A007516          (2^2^...^2) (n times) + 1.
>
>was obviously added knowing about the xref'd sequence

That isn't obvious.  We don't know when the xref was added; it may have 
come later.

>A004249         (2^2^...^2) (with n 2's) + 1.
>2,3,5,17,...
>
>and thus "wants" to be different from this,
>i.e. in A007516 the intention is NOT to think of an empty ^ tower,
>but of nothing written before the +1:
>
>For n=0, write no "2" (and consequently no "^") at all.

(Actually, you should write minus one "^"s.  I don't know how to do 
that, however.)

>Then you should see :
>+1
>
>which means, a(0) = 1.

This isn't mathematics, it's playing games with notation.  
Mathematically, A004249 is the correct sequence.

If people want to keep A007516, I have no serious objection.  I will 
continue to consider it an incorrect version of A004249, however.

>...
>%C A004249 For n=0, the empty product is meant.
>(or something similar).

It isn't the empty product, it's the empty tower.  The value of the 
empty tower is 1, for the same kind of reasons that the value of the 
empty product is 1, which is the same as why the empty sum is 
0, why 
the empty universal quanitifier is "true", why the empty existential 
quantifier is "false", why the empty union is the empty set, etc.

>...

Franklin T. Adams-Watters