[seqfan] Re: Number of points with maximum norm n in A2, A3, A4, A5, D5

[Joerg Arndt]

* Richard Mathar <mathar at strw.leidenuniv.nl> [Feb 04. 2010 10:24]:
> [...]
> 
> 5) I've tried to calculate
> "Number of lattice points in the D5 lattice with maximum norm n"
> and get for n>=0
> 1,120,1442,6840,21122,51000
> 
> [...]

looks OK:

[1, 120, 1442, 6840, 21122, 51000, 105122, 194040, 330242, 528120, 804002]


For D3 (A110907):
? Dnct(3,10)
? vc_
[1, 12, 50, 108, 194, 300, 434, 588, 770, 972, 1202]
? gf=ggf(vc_)
(x^4 + 10*x^3 + 26*x^2 + 10*x + 1)/(-x^4 + 2*x^3 - 2*x + 1)
? factor(gf)

[x^2 + 4*x + 1 1]

[x^2 + 6*x + 1 1]

[x - 1 -3]

[x + 1 -1]


Code (primitive and somewhat slow):

---------------------------------
Dnct(d,m)=
{
    if ( d<2, error("oink!") );
    vm_=vector(m);    \\ == max ( abs(x), abs(y), abs(z), ... )
    vp_=vector(m+1);  \\ parity
    vc_=vector(m+1);  \\ count
    m_ = m;
    d_ = d;
    rec(1);
} /* ----- */


rec(d)=
{
    for (x=-m_, m_,
        if ( d==1,
            vm_[d] = abs(x);
            vp_[d] = bitand(x, 1);
            rec(d+1);
        , /* else */
            vm_[d] = max(vm_[d-1], abs(x));
            vp_[d] = bitxor(vp_[d-1], bitand(x, 1));
            if ( d==d_,
                if ( vp_[d]==0,
                    vc_[vm_[d]+1] += 1;
                );
                , /* else */
                rec(d+1);
            );
        );
    );
} /* ----- */

\r ggf.gpi

default(echo,1);


Dnct(3,10)
vc_
gf=ggf(vc_)
factor(gf)
---------------------------------