莱布尼茨调和三角形
莱布尼茨调和三角形 是单位分数的三角形排列，其中最外面的对角线由行号的倒数组成，每个内部单元格是对角线上方和左侧的单元格减去左侧的单元格。用代数的方法来说， L(r，1) = 1/r ，其中 r 是从 1 开始的行数，c 是数字，不超过 r 和 L(r，c)= L(r–1，c–1)–L(r，c–1)。
与帕斯卡三角形
的关系帕斯卡三角形中的每个条目是上一行中两个条目的总和，而莱布尼茨三角形中的每个条目是下一行中两个条目的总和。例如，在第 5 行中，条目(1/30)是第 6 行中两个(1/60)的总和。
就像帕斯卡三角形可以用二项式系数计算一样，莱布尼茨也可以:
Properties
如果取第 n 行的分母并相加，那么结果将等于 n.2 n-1 。例如，对于第三行，我们有 3 + 6 + 3 = 12 = 3 × 22。
给定正整数 n 。任务是打印高度为 n 的莱布尼茨调和三角形。
示例:
Input : n = 4
Output :
1
1/2 1/2
1/3 1/6 1/3
1/4 1/12 1/12 1/4
Input : n = 3
Output :
1
1/2 1/2
1/3 1/6 1/3
下面是基于上述与帕斯卡三角形的关系，打印高度为 n 的莱布尼茨调和三角形的实现。
C++
// CPP Program to print Leibniz Harmonic Triangle
#include <bits/stdc++.h>
using namespace std;
// Print Leibniz Harmonic Triangle
void LeibnizHarmonicTriangle(int n)
{
int C[n + 1][n + 1];
// Calculate value of Binomial Coefficient in
// bottom up manner
for (int i = 0; i <= n; i++) {
for (int j = 0; j <= min(i, n); j++) {
// Base Cases
if (j == 0 || j == i)
C[i][j] = 1;
// Calculate value using previously
// stored values
else
C[i][j] = C[i - 1][j - 1] + C[i - 1][j];
}
}
// printing Leibniz Harmonic Triangle
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= i; j++)
cout << "1/" << i * C[i - 1][j - 1] << " ";
cout << endl;
}
}
// Driven Program
int main()
{
int n = 4;
LeibnizHarmonicTriangle(n);
return 0;
}
Java 语言(一种计算机语言，尤用于创建网站)
// Java Program to print
// Leibniz Harmonic Triangle
import java.io.*;
import java.math.*;
class GFG {
// Print Leibniz Harmonic Triangle
static void LeibnizHarmonicTriangle(int n)
{
int C[][] = new int[n + 1][n + 1];
// Calculate value of Binomial
// Coefficient in bottom up manner
for (int i = 0; i <= n; i++) {
for (int j = 0; j <= Math.min(i, n);
j++) {
// Base Cases
if (j == 0 || j == i)
C[i][j] = 1;
// Calculate value using
// previously stored values
else
C[i][j] = C[i - 1][j - 1] +
C[i - 1][j];
}
}
// printing Leibniz Harmonic Triangle
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= i; j++)
System.out.print("1/" + i * C[i - 1][j - 1]
+ " ");
System.out.println();
}
}
// Driven Program
public static void main(String args[])
{
int n = 4;
LeibnizHarmonicTriangle(n);
}
}
// This code is contributed by Nikita Tiwari
Python 3
# Python3 Program to print
# Leibniz Harmonic Triangle
# Print Leibniz Harmonic
# Triangle
def LeibnizHarmonicTriangle(n):
C = [[0 for x in range(n + 1)]
for y in range(n + 1)];
# Calculate value of Binomial
# Coefficient in bottom up manner
for i in range(0, n + 1):
for j in range(0, min(i, n) + 1):
# Base Cases
if (j == 0 or j == i):
C[i][j] = 1;
# Calculate value using
# previously stored values
else:
C[i][j] = (C[i - 1][j - 1] +
C[i - 1][j]);
# printing Leibniz
# Harmonic Triangle
for i in range(1, n + 1):
for j in range(1, i + 1):
print("1/", end = "");
print(i * C[i - 1][j - 1],
end = " ");
print();
# Driver Code
LeibnizHarmonicTriangle(4);
# This code is contributed
# by mits.
C
// C# Program to print Leibniz Harmonic Triangle
using System;
class GFG {
// Print Leibniz Harmonic Triangle
static void LeibnizHarmonicTriangle(int n)
{
int [,]C = new int[n + 1,n + 1];
// Calculate value of Binomial
// Coefficient in bottom up manner
for (int i = 0; i <= n; i++) {
for (int j = 0; j <= Math.Min(i, n);
j++) {
// Base Cases
if (j == 0 || j == i)
C[i,j] = 1;
// Calculate value using
// previously stored values
else
C[i,j] = C[i - 1,j - 1] +
C[i - 1,j];
}
}
// printing Leibniz Harmonic Triangle
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= i; j++)
Console.Write("1/" + i * C[i - 1,j - 1]
+ " ");
Console.WriteLine();
}
}
// Driven Program
public static void Main()
{
int n = 4;
LeibnizHarmonicTriangle(n);
}
}
// This code is contributed by vt_m.
服务器端编程语言（Professional Hypertext Preprocessor 的缩写）
<?php
// PHP Program to print
// Leibniz Harmonic Triangle
// Print Leibniz Harmonic Triangle
function LeibnizHarmonicTriangle($n)
{
// Calculate value of
// Binomial Coefficient in
// bottom up manner
for ($i = 0; $i <= $n; $i++)
{
for ($j = 0; $j <= min($i, $n); $j++)
{
// Base Cases
if ($j == 0 || $j == $i)
$C[$i][$j] = 1;
// Calculate value
// using previously
// stored values
else
$C[$i][$j] = $C[$i - 1][$j - 1] +
$C[$i - 1][$j];
}
}
// printing Leibniz
// Harmonic Triangle
for ($i = 1; $i <= $n; $i++)
{
for ($j = 1; $j <= $i; $j++)
echo "1/", $i * $C[$i - 1][$j - 1], " ";
echo "\n";
}
}
// Driver Code
$n = 4;
LeibnizHarmonicTriangle($n);
// This code is contributed by aj_36
?>
java 描述语言
<script>
// JavaScript Program to print
// Leibniz Harmonic Triangle
// Print Leibniz Harmonic Triangle
function LeibnizHarmonicTriangle(n)
{
let C = new Array(n + 1);
// Loop to create 2D array using 1D array
for (let i = 0; i < C.length; i++) {
C[i] = new Array(2);
}
// Calculate value of Binomial
// Coefficient in bottom up manner
for (let i = 0; i <= n; i++) {
for (let j = 0; j <= Math.min(i, n);
j++) {
// Base Cases
if (j == 0 || j == i)
C[i][j] = 1;
// Calculate value using
// previously stored values
else
C[i][j] = C[i - 1][j - 1] +
C[i - 1][j];
}
}
// printing Leibniz Harmonic Triangle
for (let i = 1; i <= n; i++)
{
for (let j = 1; j <= i; j++)
document.write("1/" + i * C[i - 1][j - 1]
+ " ");
document.write("<br/>");
}
}
// Driver Code
let n = 4;
LeibnizHarmonicTriangle(n);
// This code is contributed by avijitmondal1998.
</script>
输出:
1/1
1/2 1/2
1/3 1/6 1/3
1/4 1/12 1/12 1/4
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