Sequences A120588 - A120607 : Formulas?

[deutsch]

standard terminology) 
    binom(kn,n)/[(k-1)n + 1]  (n=0,1,2,...) 
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Date: Mon, 12 Nov 2007 16:43:22 -0800
From: Dean Hickerson <dean at math.ucdavis.edu>
To: seqfan at ext.jussieu.fr
Subject: Re: duplicate names
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Rainer Rosenthal wrote:

> A000446 Smallest number that is the sum of 2 squares in exactly n ways.
> A016032 Least number which is the sum of two squares in exactly n different ways.
>
> In my opinion both descriptions are equivalent. But the sequences are
> not equal and both of them lack a description.

I haven't checked all of the terms, but I think that A000446 allows 0 as a
square, and A016032 does not.  E.g. 25 = 0^2+5^2 = 3^2+4^2 is the smallest
number with exactly 2 representations if 0 is allowed, and 50 = 1^2+7^2 =
5^2+5^2 is the smallest if 0 is not allowed.

Dean Hickerson
dean at math.ucdavis.edu



It looks like A000446 treats 0 as a square and A016032 does not.

In other words A016032 knows only 1 way to represent 25 as sum of 2 squares:

25 = 9 + 16

but A000446 knows 2 ways:

25 = 0 + 25
25 = 9 + 16

And this difference is not born out in the titles.

Christian

------ Original Message ------
mathoflove-seqfan at yahoo.com

> Dear SeqFans,
> 
> could you please explain to me what the difference is between
> 
>                         exactly n ways
>             and
>                     exactly n different ways?
> 
> This question concerns OEIS sequences A000446 and A016032:
> 
> A000446 Smallest number that is the sum of 2 squares in exactly n ways.
> A016032 Least number which is the sum of two squares in exactly n different
ways.
> 
> In my opinion both descriptions are equivalent. But the sequences are
> not equal and both of them lack a description. Maybe the description
> is omitted because the title says it all? But either it doesn't or
> somebody made computing errors.
...