Markus Grassl on Mon, 04 Dec 2023 10:05:55 +0100


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Re: number of ways of writing a nonnegative integer n as a sum of 3 squares (zero being allowed).

Thanks for the clarification. I had missed that the sequence in question allows for negative integers as well.

Markus

Am 04/12/2023 um 19:09 schrieb Loïc Grenié:
On Mon Dec 4, 2023, at 08:03, Thomas D. Dean wrote:
Sorry if this too far off topic.

I have been looking at OEIS, A005875, and can not understand their
counting. I can see the Pari code and understand how it works.

OEIS calls this "number of ways of writing a nonnegative integer n as a
sum of 3 squares (zero being allowed)." or "Number of ordered triples
(i, j, k) of integers such that n = i^2 + j^2 + k^2." So, i,j,and, k
should be considered distinct even if they are numerically equal?

     No, 1 is 1 but they also consider negative integers.
 
[...]

A005875(n)=if(n<0,0,polcoeff(sum(k=1,sqrtint(n),2*x^k^2,1+x*O(x^n))^3,n))

sumsquares(n,k)=
{
    my(bd=sqrtint(n),s=[]);
    forvec(x=vector(k,i,[0,bd]),

should be forvec(x=vector(k,i,[-bd,bd]),

 
          if(n==sum(i=1,k,x[i]^2),
             s=setunion(s,[vecsort(x)]);
            );
    );
    s;
}

for(i=0,30,print(i,"\t",6*#sumsquares(i,3),"\t",A005875(i)));

If I consider each of a,b,c to be distinct even though the are not
numberically:
N [a,b,c]
0 [0,0,0] -> 6 OEIS says 1 What about ordered tripes?
1 [0,0,1] -> 6 OEIS says 6 OK
6 = number of permutations of [0,0,1] and [0,0,-1]
2 [0,1,1] -> 6 OEIS says 12 - How do they get this?
number of permutations of the 4 vectors [0,\pm1,\pm1]
3 [1,1,1] -> 6 OEIS says 8 - ?
\pm[1,1,1] both make 1
permutations of [1,1,\pm1] make 3
permutations of [1,\pm1,\pm1] make 3 [opposite of the ones above]

   The other below are similar.

4 [0,0,2] -> 6 OEIS says 6 OK
5 [0,1,2] -> 6 OEIS says 24??
6 [1,1,2] -> 6 OEIS says 24??
7 none
8 [0,2,2] -> 6 OEIS says 12
...
25 [0,0,5] [0,3,4] -> 12 OEIS says 30?
26 [0,1,5] [1,3,4] -> 12 OEIS says 72?

What am I missing? Can someone explain this to me?

    Hope this helped!

         Loïc