细谷的三角形

斐波那契三角形或Hosoya三角形是基于斐波那契数的三角形排列。每个数字是左对角线或右对角线以上两个数字的总和。前几行是：

这个三角形中的数字遵循递归关系

与斐波那契数的关系
三角形中的条目满足标识

因此，两个最外面的对角线是斐波那契数，而中间垂直线上的数字是斐波那契数的平方。三角形中的所有其他数字是两个大于1的不同斐波那契数的乘积。行总和是第一个卷积的斐波那契数。
资料来源：Stackoverflow，维基百科
给定正整数n 。任务是打印大小为n的Hosoya三角形。
例子：

Input : n = 4
Output :
1
1 1
2 1 2
3 2 2 3

Input : n = 5
Output :
1
1 1
2 1 2
3 2 2 3
5 3 4 3 5

下面是打印高度为n的Hosoya三角形的实现：

C++

// CPP Program to print Hosoya's
// triangle of height n.
#include 
using namespace std;
 
int Hosoya(int n, int m)
{
    // Base case
    if ((n == 0 && m == 0) ||
        (n == 1 && m == 0) ||
        (n == 1 && m == 1) ||
        (n == 2 && m == 1))
        return 1;
 
    // Recursive step
    if (n > m)
        return Hosoya(n - 1, m)
               + Hosoya(n - 2, m);
 
    else if (m == n)
        return Hosoya(n - 1, m - 1)
               + Hosoya(n - 2, m - 2);
 
    else
        return 0;
}
 
// Print the Hosoya triangle of height n.
void printHosoya(int n)
{
    for (int i = 0; i < n; i++) {
        for (int j = 0; j <= i; j++)
            cout << Hosoya(i, j) << " ";    
 
        cout << endl;
    }
}
 
// Driven Program
int main()
{
    int n = 5;
    printHosoya(n);
    return 0;
}

Java

// Java Program to print Hosoya's
// triangle of height n.
import java.util.*;
 
class GFG {
     
    static int Hosoya(int n, int m)
    {
        // Base case
        if ((n == 0 && m == 0) ||
            (n == 1 && m == 0) ||
            (n == 1 && m == 1) ||
            (n == 2 && m == 1))
            return 1;
      
        // Recursive step
        if (n > m)
            return Hosoya(n - 1, m)
                   + Hosoya(n - 2, m);
             
        else if (m == n)
            return Hosoya(n - 1, m - 1)
                    + Hosoya(n - 2, m - 2);
             
        else
            return 0;
    }
      
    // Print the Hosoya triangle of height n.
    static void printHosoya(int n)
    {
        for (int i = 0; i < n; i++)
        {
            for (int j = 0; j <= i; j++)
                System.out.print(Hosoya(i, j)
                                        + " ");      
      
            System.out.println("");
        }
    }
 
     
    /* Driver program to test above function */
    public static void main(String[] args)
    {
        int n = 5;
        printHosoya(n);
          
    }
}
 
// This code is contributed by  Arnav Kr. Mandal.

Python3

# Python3 code to print Hosoya's
# triangle of height n.
 
def Hosoya( n , m ):
 
    # Base case
    if ((n == 0 and m == 0) or
        (n == 1 and m == 0) or
        (n == 1 and m == 1) or
        (n == 2 and m == 1)):
                return 1
     
    # Recursive step
    if n > m:
        return Hosoya(n - 1, m)
                    + Hosoya(n - 2, m)
 
    elif m == n:
        return Hosoya(n - 1, m - 1)
                        + Hosoya(n - 2,    m - 2)
 
    else:
        return 0
         
# Print the Hosoya triangle of height n.
def printHosoya( n ):
    for i in range(n):
        for j in range(i + 1):
            print(Hosoya(i, j) , end = " ")
        print("\n", end = "")
         
# Driven Code
n = 5
printHosoya(n)
 
# This code is contributed by Sharad_Bhardwaj

C#

// C# Program to print Hosoya's
// triangle of height n.
using System;
 
class GFG {
     
    static int Hosoya(int n, int m)
    {
        // Base case
        if ((n == 0 && m == 0) ||
            (n == 1 && m == 0) ||
            (n == 1 && m == 1) ||
            (n == 2 && m == 1))
            return 1;
     
        // Recursive step
        if (n > m)
            return Hosoya(n - 1, m)
                 + Hosoya(n - 2, m);
             
        else if (m == n)
            return Hosoya(n - 1, m - 1)
                 + Hosoya(n - 2, m - 2);
             
        else
            return 0;
    }
     
    // Print the Hosoya triangle of height n.
    static void printHosoya(int n)
    {
        for (int i = 0; i < n; i++)
        {
            for (int j = 0; j <= i; j++)
                Console.Write(Hosoya(i, j)
                                        + " ");
     
            Console.WriteLine("");
        }
    }
 
     
    /* Driver program to test above function */
    public static void Main()
    {
        int n = 5;
         
        printHosoya(n);
         
    }
}
 
// This code is contributed by vt_m.

PHP

 $m)
        return Hosoya($n - 1,$m) +
               Hosoya($n - 2, $m);
 
    else if ($m == $n)
        return Hosoya($n - 1, $m - 1) +
               Hosoya($n - 2, $m - 2);
 
    else
        return 0;
}
 
// Print the Hosoya
// triangle of height n.
function printHosoya( $n)
{
    for ( $i = 0; $i < $n; $i++)
    {
        for ( $j = 0; $j <= $i; $j++)
            echo Hosoya($i, $j) , " ";
            echo "\n";
    }
}
 
// Driven Code
$n = 5;
printHosoya($n);
 
// This code is contributed by anuj_67.
?>

Javascript

 
// JavaScript program to implement
// the above approach
 
    function Hosoya(n, m)
    {
        // Base case
        if ((n == 0 && m == 0) ||
            (n == 1 && m == 0) ||
            (n == 1 && m == 1) ||
            (n == 2 && m == 1))
            return 1;
        
        // Recursive step
        if (n > m)
            return Hosoya(n - 1, m)
                   + Hosoya(n - 2, m);
               
        else if (m == n)
            return Hosoya(n - 1, m - 1)
                    + Hosoya(n - 2, m - 2);
               
        else
            return 0;
    }
        
    // Print the Hosoya triangle of height n.
    function printHosoya(n)
    {
        for (let i = 0; i < n; i++)
        {
            for (let j = 0; j <= i; j++)
                document.write(Hosoya(i, j)
                                        + " ");      
        
            document.write("<br/>");
        }
    }
 
// Driver code
        let n = 5;
        printHosoya(n);
          
         // This code is contributed by sanjoy_62.

输出 ：

1 
1 1 
2 1 2 
3 2 2 3 
5 3 4 3 5 

下面是使用动态编程打印高度为n的Hosoya三角形的实现：

C++

// CPP Program to print Hosoya's triangle of height n.
#include 
#define N 5
using namespace std;
 
// Print the Hosoya triangle of height n.
void printHosoya(int n)
{
    int dp[N][N];
    memset(dp, 0, sizeof(dp));
 
    // base case.
    dp[0][0] = dp[1][0] = dp[1][1] = 1;
 
    // For each row.
    for (int i = 2; i < n; i++) {
 
        // for each column;
        for (int j = 0; j < n; j++) {
 
            // recursive steps.
            if (i > j)
                dp[i][j] = dp[i - 1][j] + dp[i - 2][j];
 
            else
                dp[i][j] = dp[i - 1][j - 1] + dp[i - 2][j - 2];
        }
    }
 
    // printing the solution
    for (int i = 0; i < n; i++) {
        for (int j = 0; j <= i; j++)
            cout << dp[i][j] << " ";       
 
        cout << endl;
    }
}
 
// Driven Program
int main()
{
    int n = 5;
    printHosoya(n);
    return 0;
}

Java

// JAVA Code for Hosoya Triangle
import java.util.*;
 
class GFG {
     
    static int N = 5;
     
    // Print the Hosoya triangle
    // of height n.
    static void printHosoya(int n)
    {
        int dp[][] = new int[N][N];
         
        // base case.
        dp[0][0] = dp[1][0] = 1;
        dp[1][1] = 1;
      
        // For each row.
        for (int i = 2; i < n; i++)
        {
            // for each column;
            for (int j = 0; j < n; j++)
            {
                 // recursive steps.
                if (i > j)
                    dp[i][j] = dp[i - 1][j] +
                                        dp[i - 2][j];
      
                else
                    dp[i][j] = dp[i - 1][j - 1] +
                                    dp[i - 2][j - 2];
            }
        }
      
        // printing the solution
        for (int i = 0; i < n; i++)
        {
            for (int j = 0; j <= i; j++)
                System.out.print(dp[i][j] + " ");       
      
            System.out.println("");
        }
    }
     
    /* Driver program*/
    public static void main(String[] args)
    {
        int n = 5;
        printHosoya(n);
    }
}
 
// This code is contributed by Arnav Kr. Mandal.

Python3

# Python3 Program to print
# Hosoya's triangle of height n.
N = 5
 
# Print the Hosoya triangle
# of height n.
def printHosoya(n):
    dp = [[0 for i in range(N)]
             for i in range(N)]
              
    # base case.
    dp[0][0] = dp[1][0] = dp[1][1] = 1
     
    # For each row.
    for i in range(2, n):
         
        # for each column
        for j in range(n):
             
            # recursive steps.
            if (i > j):
                dp[i][j] = (dp[i - 1][j] +
                            dp[i - 2][j])
            else:
                dp[i][j] = (dp[i - 1][j - 1] +
                            dp[i - 2][j - 2])
                             
    # printing the solution
    for i in range(n):
        for j in range(i + 1):
            print(dp[i][j], end = ' ')
        print()
 
# Driver Code
n = 5
printHosoya(n)
 
# This code is contributed
# by sahilshelangia

C＃

// C# Code for Hosoya Triangle
using System;
 
class GFG {
     
    static int N = 5;
     
    // Print the Hosoya triangle
    // of height n.
    static void printHosoya(int n)
    {
        int [,]dp = new int[N,N];
         
        // base case.
        dp[0,0] = dp[1,0] = 1;
        dp[1,1] = 1;
     
        // For each row.
        for (int i = 2; i < n; i++)
        {
            // for each column;
            for (int j = 0; j < n; j++)
            {
                // recursive steps.
                if (i > j)
                    dp[i,j] = dp[i - 1,j] +
                              dp[i - 2,j];
     
                else
                    dp[i,j] = dp[i - 1,j - 1]
                           + dp[i - 2,j - 2];
            }
        }
     
        // printing the solution
        for (int i = 0; i < n; i++)
        {
            for (int j = 0; j <= i; j++)
                Console.Write(dp[i,j] + " ");
     
            Console.WriteLine("");
        }
    }
     
    /* Driver program*/
    public static void Main()
    {
        int n = 5;
         
        printHosoya(n);
    }
}
 
// This code is contributed by Vt_m.

的PHP

 $j)
                $dp[$i][$j] = $dp[$i - 1][$j]
                            + $dp[$i - 2][$j];
 
            else
                $dp[$i][$j] = $dp[$i - 1][$j - 1]
                            + $dp[$i - 2][$j - 2];
        }
    }
 
    // printing the solution
    for ($i = 0; $i < $n; $i++) {
        for ($j = 0; $j <= $i; $j++)
            echo $dp[$i][$j]." ";
 
        echo "\n";
    }
}
 
// Driven Program
 
    $n = 5;
    printHosoya($n);
 
// This code is contributed by mits
?>

输出 ：

1 
1 1 
2 1 2 
3 2 2 3 
5 3 4 3 5