[seqfan] integer lattice points in balls of higher dimensions

[Richard Mathar]

We have counts of integer lattice points in spheres of dimension
d=2 (A000328), d=3 (A000605), d=4 (A055410), d=5 (A055411) etc.

It seems that the equivalent count of solutions for d-tuples (x_1,x_2,...,x_d)
  sum_{j=1..d}  x_j^p <= n^2 (offset for n is 0)
for powers (''metric'') p>2 are not yet in the OEIS (?)
I get for example by brute force scanning of all -n <= x_j <= n:

d=2,p=3:
1,8,18,37,55,78,111,142,184,225,272,319,379,438,499,574,,642,717,794,885,1062

d=2,p=4:
1,5,9,9,13,21,25,25,25,29,45,45,45,49,49,49,53,69,69,77,77,77,77,81,81,85

d=2,p=5:
1,8,18,31,48,69,100,129,163,200,241,286,335,388,445,508,579,650,723

d=2,p=6:
1,5,9,9,9,9,9,9,13,21,21,21,25,25,25,25,25,25,25,25,25


d=3,p=3:
1,23,85,236,459,803,1347,1990,2915,3967,5321,6873,8827,11027,13558,16571

d=3,p=4:
1,7,27,27,33,81,117,125,125,131,251,275,275,311,335,335

d=3,p=5:
1,23,85,203,405,715,1211,1815,2605,3596,4813,6281,8025,10069,12440,15174

d=3,p=6:
1,7,27,27,27,27,27,27,33,81,81,81,117,117,125,125,125,125,125,125,125

Perhaps some of the first differences are in the OEIS (?); looking at this
as peeling an onion in higher dimensions..
Do I miss something here? Is anyone eager to submit these?

In Maple:

GaussCProp := proc(n,p,d)
        local a,x,y,z,k,l ;
        a := 0 ;
        if d = 2 then
                for x from -n to n do
                for y from -n to n do
                        if x^p+y^p <= n^2 then
                                a := a+1 ;
                        end if;
                end do:
                end do:
        elif d = 3 then
                for x from -n to n do
                for y from -n to n do
                for z from -n to n do
                        if x^p+y^p+z^p <= n^2 then
                                a := a+1 ;
                        end if;
                end do:
                end do:
                end do:
        elif d = 4 then
                for x from -n to n do
                for y from -n to n do
                for z from -n to n do
                for k from -n to n do
                        if x^p+y^p +z^p+k^p <= n^2 then
                                a := a+1 ;
                        end if;
                end do:
                end do:
                end do:
                end do:
        elif d = 5 then
                for x from -n to n do
                for y from -n to n do
                for z from -n to n do
                for k from -n to n do
                for l from -n to n do
                        if x^p+y^p +z^p+k^p+l^p <= n^2 then
                                a := a+1 ;
                        end if;
                end do:
                end do:
                end do:
                end do:
                end do:
        end if;
        a ;
end proc:
seq(GaussCProp(n,6,3),n=0..20) ;