信奥必学之卡特兰数李兴球
教学我就不提供了,这里首先提供Python代码:
# 假设一只昆虫从(0,0)要爬到(n,n)，但是只能向右爬或者向上爬，并且不能越过对角线。
# 那么会有多少种方案呢？
作者 = '李兴球'
日期 = '2025/3/26'
path = []
counter = 0
def dfs(x,y):
global counter
if x==n and y==n:
counter += 1
else:
for dx,dy in [(1,0),(0,1)]:#只两个方向
next_x = x + dx
next_y = y + dy
if next_y>next_x:continue
if next_x>n or next_y>n:continue
path.append((next_x,next_y))
dfs(next_x,next_y)
path.pop()
for n in range(10):
path.append((0,0))
counter = 0
dfs(0,0)
print(counter,end=',')
然后是C++的代码:
/*
作者：李兴球
日期：2025/3/26
*/
#include
#include
#define pos pair<int,int>
using namespace std;
vector< pos > path;
int dirs[2][2] = {{1,0},{0,1}}; //两个方向
int n,ans;
void dfs(int x,int y){
if(x==n && y==n)
ans++;
//for(pos p:path) cout << p.first << ',' << p.second << '->';
//cout << endl; return;
for(int fx=0;fx<2;fx++){
int next_x = x + dirs[fx][0];
int next_y = y + dirs[fx][1];
if(next_y>next_x || next_x>n || next_y>n)continue;
path.push_back({next_x,next_y});
dfs(next_x,next_y);
path.pop_back();
}
}
int main(){
for( n=0;n<10;n++){
ans=0; path.clear();
path.push_back({0,0});
dfs(0,0);
cout << ans << ' ';
}
}
最后就是用Python做的动画,你问我列表中的坐标从哪里来,还不是从上面的程序中来的:
from sprites import *
bug = Sprite()
bug.pencolor('blue')
bug.pensize(4)
length= 80
cors = bug.draw_grid2(6,6,length,length)
left,bottom = cors[5][0]
left -= length//2
bottom -= length//2
right,top = cors[0][5]
right += length//2
top += length//2
bug.goto(left,bottom)
bug.pendown()
bug.bk(50);bug.fd(600);bug.shape('classic');bug.stamp()
bug.bk(550);bug.left(90);bug.fd(600);bug.stamp();bug.bk(600)
bug.shape('res/bug.png')
bug.pensize(2)
bug.pencolor('magenta')
bug.dot(20,'magenta')
bug.draw_line((left,bottom),(right,top),True,10)
bug.dot(20,'magenta')
all_paths = [
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (5, 0), (6, 0), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (5, 0), (5, 1), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (5, 0), (5, 1), (5, 2), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (5, 0), (5, 1), (5, 2), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (5, 0), (5, 1), (5, 2), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (5, 0), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (4, 1), (5, 1), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (4, 1), (5, 1), (5, 2), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (4, 1), (5, 1), (5, 2), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (4, 1), (5, 1), (5, 2), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (4, 1), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (4, 1), (4, 2), (5, 2), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (4, 1), (4, 2), (5, 2), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (4, 1), (4, 2), (5, 2), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (4, 1), (4, 2), (5, 2), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (4, 1), (4, 2), (4, 3), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (4, 1), (4, 2), (4, 3), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (4, 1), (4, 2), (4, 3), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (4, 1), (4, 2), (4, 3), (4, 4), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (4, 1), (4, 2), (4, 3), (4, 4), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (4, 1), (5, 1), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (4, 1), (5, 1), (5, 2), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (4, 1), (5, 1), (5, 2), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (4, 1), (5, 1), (5, 2), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (4, 1), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (4, 1), (4, 2), (5, 2), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (4, 1), (4, 2), (5, 2), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (4, 1), (4, 2), (5, 2), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (4, 1), (4, 2), (5, 2), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (4, 1), (4, 2), (4, 3), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (4, 1), (4, 2), (4, 3), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (4, 1), (4, 2), (4, 3), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (4, 1), (4, 2), (4, 3), (4, 4), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (4, 1), (4, 2), (4, 3), (4, 4), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (3, 2), (4, 2), (5, 2), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (3, 2), (4, 2), (5, 2), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (3, 2), (4, 2), (5, 2), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (3, 2), (4, 2), (5, 2), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (3, 2), (4, 2), (4, 3), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (3, 2), (4, 2), (4, 3), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (3, 2), (4, 2), (4, 3), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (3, 2), (4, 2), (4, 3), (4, 4), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (3, 2), (4, 2), (4, 3), (4, 4), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (3, 2), (3, 3), (4, 3), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (3, 2), (3, 3), (4, 3), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (3, 2), (3, 3), (4, 3), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (3, 2), (3, 3), (4, 3), (4, 4), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (3, 2), (3, 3), (4, 3), (4, 4), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (4, 1), (5, 1), (5, 2), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (4, 1), (5, 1), (5, 2), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (4, 1), (5, 1), (5, 2), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (4, 1), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (4, 1), (4, 2), (5, 2), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (4, 1), (4, 2), (5, 2), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (4, 1), (4, 2), (5, 2), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (4, 1), (4, 2), (5, 2), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (4, 1), (4, 2), (4, 3), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (4, 1), (4, 2), (4, 3), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (4, 1), (4, 2), (4, 3), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (4, 1), (4, 2), (4, 3), (4, 4), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (4, 1), (4, 2), (4, 3), (4, 4), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (3, 2), (4, 2), (5, 2), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (3, 2), (4, 2), (5, 2), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (3, 2), (4, 2), (5, 2), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (3, 2), (4, 2), (5, 2), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (3, 2), (4, 2), (4, 3), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (3, 2), (4, 2), (4, 3), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (3, 2), (4, 2), (4, 3), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (3, 2), (4, 2), (4, 3), (4, 4), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (3, 2), (4, 2), (4, 3), (4, 4), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (3, 2), (3, 3), (4, 3), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (3, 2), (3, 3), (4, 3), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (3, 2), (3, 3), (4, 3), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (3, 2), (3, 3), (4, 3), (4, 4), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (3, 1), (3, 2), (3, 3), (4, 3), (4, 4), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (2, 2), (3, 2), (4, 2), (5, 2), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (2, 2), (3, 2), (4, 2), (5, 2), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (2, 2), (3, 2), (4, 2), (5, 2), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (2, 2), (3, 2), (4, 2), (4, 3), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (2, 2), (3, 2), (4, 2), (4, 3), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (2, 2), (3, 2), (4, 2), (4, 3), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (2, 2), (3, 2), (4, 2), (4, 3), (4, 4), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (2, 2), (3, 2), (4, 2), (4, 3), (4, 4), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (2, 2), (3, 2), (3, 3), (4, 3), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (2, 2), (3, 2), (3, 3), (4, 3), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (2, 2), (3, 2), (3, 3), (4, 3), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (2, 2), (3, 2), (3, 3), (4, 3), (4, 4), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (2, 0), (2, 1), (2, 2), (3, 2), (3, 3), (4, 3), (4, 4), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (5, 2), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (5, 2), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (5, 2), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (4, 1), (4, 2), (5, 2), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (4, 1), (4, 2), (5, 2), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (4, 1), (4, 2), (5, 2), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (4, 1), (4, 2), (5, 2), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (4, 1), (4, 2), (4, 3), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (4, 1), (4, 2), (4, 3), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (4, 1), (4, 2), (4, 3), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (4, 1), (4, 2), (4, 3), (4, 4), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (4, 1), (4, 2), (4, 3), (4, 4), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (3, 2), (4, 2), (5, 2), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (3, 2), (4, 2), (5, 2), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (3, 2), (4, 2), (5, 2), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (3, 2), (4, 2), (5, 2), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (3, 2), (4, 2), (4, 3), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (3, 2), (4, 2), (4, 3), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (3, 2), (4, 2), (4, 3), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (3, 2), (4, 2), (4, 3), (4, 4), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (3, 2), (4, 2), (4, 3), (4, 4), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (3, 2), (3, 3), (4, 3), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (3, 2), (3, 3), (4, 3), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (3, 2), (3, 3), (4, 3), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (3, 2), (3, 3), (4, 3), (4, 4), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (3, 2), (3, 3), (4, 3), (4, 4), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (2, 2), (3, 2), (4, 2), (5, 2), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (2, 2), (3, 2), (4, 2), (5, 2), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (2, 2), (3, 2), (4, 2), (5, 2), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (2, 2), (3, 2), (4, 2), (4, 3), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (2, 2), (3, 2), (4, 2), (4, 3), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (2, 2), (3, 2), (4, 2), (4, 3), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (2, 2), (3, 2), (4, 2), (4, 3), (4, 4), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (2, 2), (3, 2), (4, 2), (4, 3), (4, 4), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (2, 2), (3, 2), (3, 3), (4, 3), (5, 3), (6, 3), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (2, 2), (3, 2), (3, 3), (4, 3), (5, 3), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (2, 2), (3, 2), (3, 3), (4, 3), (5, 3), (5, 4), (5, 5), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (2, 2), (3, 2), (3, 3), (4, 3), (4, 4), (5, 4), (6, 4), (6, 5), (6, 6)],
[(0, 0), (1, 0), (1, 1), (2, 1), (2, 2), (3, 2), (3, 3), (4, 3), (4, 4), (5, 4), (5, 5), (6, 5), (6, 6)] ]
bug.hide()
bug2 = Sprite()
bug2.color('red')
bug2.pensize(8)
x0 = left
y0 = bottom
for path in all_paths:
bug2.goto(x0,y0)
bug2.pendown()
bug2.setheading(0)
bug2.wait(0.5)
bug2.clear()
#下面就是相对前进
for i in range(len(path)-1):
dx = (path[i+1][0] - path[i][0])* length
dy = (path[i+1][1] - path[i][1])* length
if dx==0: # 则肯定是dy不是0，则往上走
bug2.setheading(90)
bug2.fd(length)
else:
bug2.setheading(0)
bug2.fd(length)
bug2.wait(0.1)
bug2.wait(0.5)
bug2.penup()
bug2.goto(x0,y0)
根据组合原理,卡特兰数的通项公式是:
第n项是: C(2n,n)/(n+1)
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输入 katelandfs 可下载演示视频关于卡特兰数的Python,C++和动画制作所有源代码
视频演示地址:
https://mp.weixin.qq.com/cgi-bin/home?t=home/index&token=1109013377&lang=zh_CN
