Help needed with new sequences

[Gerald McGarvey]

For n = x^2 + y^2 + T_z, if x,y,z must be non-negative and order matters, 
this is what I get for n=0 to 55:

1,3,3,2,4,5,3,4,4,3,7,7,3,4,5,5,8,5,4,9,8,4,4,8,3,8,12,4,8,7,7,8,9,4,4,11,6,12,8,5,12,12,5,4,9,7,15,8,4,8,11,9,8,15,7,13 


To get this I used Excel, calculating the sum for all combinations of x,y, 
and z from 0 to 10,
converting the formulas to values, sorting, using the frequency function, 
then truncating the
results at n=55. Not a rigorous approach and needs verification, but it 
could be used as a check.
The values seem to go up logarithmically, roughly speaking. For 2<n<=55,
a(n)/ln(n) is bounded between 3.9178... and .94397...

-- Gerald

At 07:33 PM 5/13/2005, N. J. A. Sloane wrote:

>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