Help needed with new sequences

[David Wilson]

There are several conceivable ways to count solutions of S(a)+S(b)+T(c) = n.
The counts depend on what we recognize as distinct solutions.

For example, we might consider S(a)+S(b)+S(c) equivalent to 
S(a')+S(b')+S(c') if

1.  Ordered triple (S(a), S(b), S(c)) = (S(a'), S(b'), S(c')).  In this 
case, S(0)+S(1)+T(2) is counted as distinct from S(1)+S(0)+S(2) because 
corresponding term values differ.

2.  ({S(a), S(b)}, S(c)) = ({S(a'), S(b')}, S(c')).  In this case, 
S(0)+S(1)+T(2) is counted as equivalent to S(1)+S(0)+T(2), that is, 
different orders of adding the squares are considered equivalent.

3.  Multiset [S(a), S(b), S(c)] = [S(a'), S(b'), S(c')].  In this case 
S(1)+S(6)+T(8) is counted as equivalent to S(6)+S(6)+T(1), since they both 
represent 1+36+36.  Only the values of the terms matters.

I'm sure there are others.

----- Original Message ----- 
From: "wouter meeussen" <wouter.meeussen at pandora.be>
To: "Seqfan (E-mail)" <seqfan at ext.jussieu.fr>
Sent: Saturday, May 14, 2005 6:08 AM
Subject: Re: Help needed with new sequences


> sorry if I don't get it, but
> since
> Sum[x^(k(k + 1)/2), {k, 0, \[Infinity]}]
> =EllipticTheta[2, 0, x]/(2 x^(1/8))
>
> and
> Sum[x^(k*k), {k, 0, \[Infinity]}]
> =1/2*(1 + EllipticTheta[3, 0, x])
>
> it is easy to write the GF for this:
>
> I find
> CoefficientList[Series[EllipticTheta[2, 0, Sqrt[x]]/(2*x^(1/8))*1/2*
>    (1 + EllipticTheta[3, 0, x])*1/2*(1 + EllipticTheta[3, 0, x]), {x, 0, 
> 64}], x]
> =
> {1, 3, 3, 2, 4, 5, 3, 4, 4, 3, 7, 7, 3, 4, 5, 5, 8, 5, 4, 9, 8, 4, 4, 8, 
> 3,
> 8, 12, 4, 8, 7, 7, 8, 9, 4, 4, 11, 5, 10, 7, 5, 10, 10, 5, 4, 8, 4, 11, 7, 
> 4,
> 4, 9, 9, 6, 12, 4, 10, 7, 4, 6, 6, 6, 6, 8, 2, 10}
>
> Am I being silly here?
>
> W.
>
> ----- Original Message ----- 
> From: "N. J. A. Sloane" <njas at research.att.com>
> To: <seqfan at ext.jussieu.fr>
> Cc: <njas at research.att.com>
> Sent: Saturday, May 14, 2005 1:33 AM
> Subject: Help needed with new sequences
>
>
>
> Dear Seqfans,  Professor Zhi-Wei Sun and his students have been looking
> at numbers that can be written (for example) in the form x^2 + y^2 + T_z,
> and they show that every natural number n can be written in this form.
> (T_z = z(z+1)/2)
>
> So it is natural to ask, how many ways are there of writing
> n in the form x^2 + y^2 + T_z?  (S+S+T, for short)
>
> They consider MANY other mixed sums of squares (S) and triangular numbers 
> (T)
> (such as S+2S+T, S+2T+T, ...)
>
> There are two papers on the arXiv:
> http://front.math.ucdavis.edu/math.NT/0505128
> http://front.math.ucdavis.edu/math.NT/0505187
> and possibly more on his home page:
> http://pweb.nju.edu.cn/zwsun
>
> For each of these we may ask, how many ways are possible?
> And there can be several answers, depending on
> whether order or signs are taken into account.
>
> Sequence A005875 is the classical sequence that gives the number
> of ways of writing n as a sum of 3 squares, taking
> order and signs into account.
> But if you ignore signs and order you
> get a sequence which begins (I think) 1,1,1,1,1,1,0,1,2,1,...,
> -I think David Wilson would call the latter sequence
> "Number of ways to partition n into 3 or fewer squares"
> (I didn't stop to find out the A-number).
>
> So here are a lot of potential new sequences from Sun's papers,
> if one or more seqfans would like to compute them!
> There is enough material for a collaborative effort.
> If you are going to work on this, post a note here.  Thanks!
> NJAS
>
>
>