Bill Allombert on Mon, 04 Dec 2023 11:35:24 +0100

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

 To: pariusers@pari.math.ubordeaux.fr
 Subject: Re: number of ways of writing a nonnegative integer n as a sum of 3 squares (zero being allowed).
 From: Bill Allombert <Bill.Allombert@math.ubordeaux.fr>
 Date: Mon, 4 Dec 2023 10:17:20 +0100
 Deliverydate: Mon, 04 Dec 2023 11:35:24 +0100
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On Sun, Dec 03, 2023 at 11:03:08PM 0800, 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?
I can only assume that by 'ordered triples' they just means vectors of length 3,
and not that the values are ordered.
Try this:
fun(n)=my(a=sqrtint(n),c=0);for(i=a,a,for(j=a,a,for(k=a,a,if(i^2+j^2+k^2==n,print(c++,":",[i,j,k])))));
? fun(1)
1:[1,0,0]
2:[0,1,0]
3:[0,0,1]
4:[0,0,1]
5:[0,1,0]
6:[1,0,0]
? fun(2)
1:[1,1,0]
2:[1,0,1]
3:[1,0,1]
4:[1,1,0]
5:[0,1,1]
6:[0,1,1]
7:[0,1,1]
8:[0,1,1]
9:[1,1,0]
10:[1,0,1]
11:[1,0,1]
12:[1,1,0]
etc.
Cheers,
Bill.