细谷三角

斐波那契三角形或细谷三角形是基于斐波那契数列的数字三角形排列。每个数字是左对角线或右对角线上的两个数字之和。前几行是：

这个三角形中的数字遵循递推关系

与斐波那契数列的关系
三角形中的条目满足身份

因此，最外面的两条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都是两个不同的大于 1 的斐波那契数的乘积。行和是第一个卷积的斐波那契数。
资料来源：Stackoverflow，维基百科
给定一个正整数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

下面是打印高为 n 的细谷三角形的实现：

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 

下面是使用动态规划打印高为 n 的细谷三角形的实现：

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.

蟒蛇3

# 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.

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
?>

Javascript

 
// Javascript Program to print Hosoya's
// triangle of height n.
var N = 5
 
// Print the Hosoya triangle of height n.
function printHosoya(n)
{
    var dp = Array.from(Array(N),
    ()=> Array(N).fill(0));
 
    // base case.
    dp[0][0] = dp[1][0] = dp[1][1] = 1;
 
    // For each row.
    for (var i = 2; i < n; i++) {
 
        // for each column;
        for (var 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 (var i = 0; i < n; i++) {
        for (var j = 0; j <= i; j++)
            document.write( dp[i][j] + " ");       
 
        document.write("<br>");
    }
}
 
// Driven Program
var n = 5;
printHosoya(n);
 
   

输出：

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