提供參考
我的作法是慢慢加大盒子大小
每次增加盒子大小就把「座標點」寫到陣列（可排序後輸出）

座標點的表達我是規定從大排到小
例如：(5,5,2) 這組數字，代表了 個座標點

#include <iostream>
#include <vector>
#include <algorithm>

class Point
{
private:
    unsigned x, y, z; //座標（限定 x >= y >= z >= 0）
    unsigned step;    //步長(奇偶數決定正負號)
    unsigned r2;      //半徑平方
    unsigned n;       //對稱點數目
    bool xAxis;       //是否在 x 軸上

public:
    Point(unsigned, unsigned, unsigned); //建構，參數請從大排到小
    bool isXAxis();                      //這個點是否在 x 軸上
    unsigned _getNum();                  //取得對稱點數目
    bool operator<(const Point &);       //用於 std::sort
    friend std::ostream &operator<<(std::ostream &, const Point &); //輸出
};

class BoxPoints
{
private:
    unsigned r;   //盒子中心到表面的最短距離
    unsigned idx; //讀取到的位置
    std::vector<Point> points; //收集到的點
    void _extendRadius(); //加大盒子大小，更新 r、points、idx

public:
    BoxPoints();
    Point getNext(); //取得下一個點
};

int main()
{
    BoxPoints box;
    for (int i = 0; i < 20; ++i)
    {
        std::cout << box.getNext();
    }
    return 0;
}

Point::Point(unsigned px, unsigned py, unsigned pz)
{
    x = px;
    y = py;
    z = pz;
    step = px + py + pz;
    r2 = px * px + py * py + pz * pz;
    n = _getNum();
    xAxis = (y == 0 && z == 0);
}

bool Point::isXAxis()
{
    return xAxis;
}

unsigned Point::_getNum()
{
    unsigned t = 1;
    if (x > 0)
    {
        t <<= 1;
    }
    if (y > 0)
    {
        t <<= 1;
    }
    if (z > 0)
    {
        t <<= 1;
    }
    if (x == y)
    {
        if (y == z)
        {
            return t;
        }
        else
        {
            return t * 3;
        }
    }
    else
    {
        if (y == z)
        {
            return t * 3;
        }
        else
        {
            return t * 6;
        }
    }
}

bool Point::operator<(const Point &p)
{
    return r2 != p.r2 ? (r2 < p.r2) : (step < p.step);
}

std::ostream &operator<<(std::ostream &os, const Point &p)
{
    os << "(" << p.x << "," << p.y << "," << p.z << ") => "
       << (p.step % 2 ? "-" : "+") << p.n << "/" << p.r2 << " (" << p.step << ")\n";
    return os;
}

BoxPoints::BoxPoints()
{
    r = 0;
    idx = 0; //讀取的索引
    _extendRadius();
}

Point BoxPoints::getNext()
{
    Point p = points[idx++];
    if (p.isXAxis())
    {
        points.erase(points.begin(), points.begin() + idx);
        idx = 0;
        _extendRadius();
    }
    return p;
}

void BoxPoints::_extendRadius()
{
    unsigned x = ++r;
    for (int y = x; y >= 0; --y)
    {
        for (int z = y; z >= 0; --z)
        {
            points.push_back(Point(x, y, z));
        }
    }
    std::sort(points.begin(), points.end());
}

/*
輸出結果
(1,0,0) => -6/1 (1)
(1,1,0) => +12/2 (2)
(1,1,1) => -8/3 (3)
(2,0,0) => +6/4 (2)
(2,1,0) => -24/5 (3)
(2,1,1) => +24/6 (4)
(2,2,0) => +12/8 (4)
(3,0,0) => -6/9 (3)
(2,2,1) => -24/9 (5)
(3,1,0) => +24/10 (4)
(3,1,1) => -24/11 (5)
(2,2,2) => +8/12 (6)
(3,2,0) => -24/13 (5)
(3,2,1) => +48/14 (6)
(4,0,0) => +6/16 (4)
(4,1,0) => -24/17 (5)
(3,2,2) => -24/17 (7)
(3,3,0) => +12/18 (6)
(4,1,1) => +24/18 (6)
(3,3,1) => -24/19 (7)
*/

修改後

#include <iostream>
#include <cmath>
using namespace std;

int main(void){
    int a,b,h,i,j,k,N=0;
    char sign;
    cin >> a >> b;
    for(h=a;h<=b;h++){
        if(h%2==1){
            sign='-';
        }
        else{
            sign='+';
        }
        for(i=-h;i<=h;i++){
            for(j=-h;j<=h;j++){
                for(k=-h;k<=h;k++){
                    if(((i*i)+(j*j)+(k*k))==(h*h)){
                        N=N+1;
                    }
                }
            }
        }
        cout << h << "\t" << sign << "\t" << N << "\n";
        N=0;
    }
    return 0;
}