Re: question on class number for a certain n

Randall, thanks for the positive feedback.

I've polished and published my script at
https://oeis.org/wiki/User:Max_Alekseyev/gpscripts#Number_of_Partitions%2FCompositions_into_Integer_Squares
Please note that function p3(n) is now called p3_nn(n).

Regards,
Max

On Wed, Nov 19, 2025 at 6:24 PM American Citizen <[email protected]>
wrote:

> Max:
>
> I checked using your p3(n) function, and it indeed matches my results for
> the first 237 relative maximums as N gradually climbs in value.
>
> Thank you for posting this
>
> Randall
> On 11/19/25 14:37, Max Alekseyev wrote:
>
> Function p3(n) in the attached code works for all n.
>
> Regards,
> Max
>
>
>
> On Wed, Nov 19, 2025 at 3:24 PM Bill Allombert <
> [email protected]> wrote:
>
>> On Wed, Nov 19, 2025 at 10:16:28AM +0100, Bill Allombert wrote:
>> > On Tue, Nov 18, 2025 at 10:59:53PM -0800, American Citizen wrote:
>> > > Hi all:
>> > >
>> > > I have been looking at representations of integers as the sum of 3
>> squares
>> > > and things are very interesting.
>> > >
>> > > A quote from Wolfram Math states
>> > >
>> > > > The number of solutions of
>> > > >
>> > > > (36)   x^2 + y^2 + z^2 = n
>> > > >
>> > > > for a given n without restriction on the signs or relative sizes of
>> x,
>> > > > y, and z is given by r_3(n). Gauss proved that if n is squarefree
>> and
>> > > > n>4, then
>> > > >
>> > > > (37)  r_3(n) =  24h(-n)  for n=3 (mod 8);
>> > > >              =  12h(-4n) for n=1,2,5,6 (mod 8);
>> > > >              =   0       for n=7 (mod 8)
>> > > >
>> > > > (Arno 1992), where h(x) is the class number of x.
>> > > subsequent post. Can r_3(n) be found?
>> >
>> > This is a classical formula, but beware, it count all ordered triples
>> (x,y,z)
>> > in Z^3, so for example for n=17 there are 48 solutions instead of 2.
>>
>> I have found a formula (also for n>4 squarefree)
>> that only count positive increasing triples 0<=a<=b<=c.
>>
>> The idea is to count 'exceptional' representations separately, that is
>> the representations
>> that include repeated terms or the number 0, and use linear algebra.
>>
>> ? cnt(416666)
>> %2 = 339
>>
>> Cheers,
>> Bill
>>
>