斐波那契系数和斐波那契三角形

斐波那契系数
在数学中，斐波那契系数或斐波那契二项式系数定义为

$\binom{n}{k}_f = \frac{F_n F_{n-1}....F_{n-k+1} }{F_k F_{k-1}....F1} = \frac{n_F!}{k_F!(n-k)_F!}$
其中n和k是非负整数，0≤k≤n，F _j是第j个斐波那契数，n！ _F是第n个Fibonorial，其中0！ _F是空产品，其值为1。
斐波那契系数都是整数。一些特殊的值是：
$\binom{n}{0}_f = \binom{n}{n}_f = 1$
$\binom{n}{1}_f = \binom{n}{n-1}_f = F_n$
$\binom{n}{2}_f = \binom{n}{n-2}_f = \frac{F_nF_{n-1}}{F_2F_1}$
$\binom{n}{3}_f = \binom{n}{n-3}_f = \frac{F_nF_{n-1}F_{n-2}}{F_3F_2F_1}$
$\binom{n}{k}_f = \binom{n}{n-k}_f$

斐波那契三角形
斐波那契系数类似于二项式系数，可以显示为类似于Pascal三角形的三角形。前八行如下所示。

斐波那契三角形的递归关系：
$\binom{n}{k}_f = F_{n-k+1} \binom{n-1}{k-1}_f + F_{k-1} \binom{n-1}{k}_f$
给定正整数n 。任务是打印高度为n(或n + 1行)的斐波那契三角形。

例子：

Input : n = 6
Output :
1
1 1
1 1 1
1 2 2 1
1 3 6 3 1
1 5 15 15 5 1
1 8 40 60 40 8 1

Input : n = 5
Output :
1
1 1
1 1 1
1 2 2 1
1 3 6 3 1
1 5 15 15 5 1

下面是打印高度为n的斐波那契三角形的实现：

C++

// CPP Program to print Fibonomial Triangle of height n.
#include
#define N 6
using namespace std;
  
// Function to produce Fibonacci Series.
void fib(int f[], int n)
{
    int i;
   
    /* 0th and 1st number of the series are 0 and 1*/
    f[0] = 0;
    f[1] = 1;
   
    for (i = 2; i <= n; i++)
      
        /* Add the previous 2 numbers in the series
         and store it */
        f[i] = f[i-1] + f[i-2];    
}
  
// Function to produce fibonomial coefficient
void fibcoef(int fc[][N+1], int f[], int n)
{
    for (int i = 0; i <= n; i++)
        fc[i][0] = 1;
          
    for (int i = 1; i <= n; i++)
    {
        for (int j = 1; j <= i; j++)
        {
            int k = j;
            while(k--)
                fc[i][j] *= f[k];
                  
            k = 1;
              
            while((j+1)!=k)
                fc[i][j] /= f[k++];
        }
    }
}
  
// Function to print Fibonomial Triangle.
void printFibonomialTriangle(int n)
{
    int f[N+1] = { 0 };
      
    // Finding the fibonacci series.
    fib(f, n);
      
    // to store triangle value.
    int dp[N+1][N+1] = { 0 };
      
    // initalising the 0th element of each row 
    // and diagonal element equal to 0.
    for (int i = 0; i <= n; i++)
        dp[i][0] = dp[i][i] = 1;
          
    // for each row.
    for (int i = 1; i <= n; i++)
    {
        // for each column.
        for (int j = 1; j < i; j++)
          
            // finding each element using recurrence 
            // relation.
            dp[i][j] = f[i-j+1]*dp[i-1][j-1] +
                       f[j-1]*dp[i-1][j];    
    }
      
    // printing the Fibonomial Triangle.
    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 = 6;    
    printFibonomialTriangle(n);
    return 0;
}

Java

// Java Program to print Fibonomial 
// Triangle of height n.
class GFG
{
    static final int N=6;
      
    // Function to produce Fibonacci Series.
    static void fib(int f[], int n)
    {
        int i;
      
        /* 0th and 1st number of
         the series are 0 and 1*/
        f[0] = 0;
        f[1] = 1;
      
        for (i = 2; i <= n; i++)
          
            /* Add the previous 2 numbers in 
            the series and store it */
            f[i] = f[i-1] + f[i-2]; 
    }
      
    // Function to produce fibonomial coefficient
    static void fibcoef(int fc[][], int f[], int n)
    {
        for (int i = 0; i <= n; i++)
            fc[i][0] = 1;
              
        for (int i = 1; i <= n; i++)
        {
            for (int j = 1; j <= i; j++)
            {
                int k = j;
                  
                while(k > 0)
                {
                    k--;
                    fc[i][j] *= f[k];
                }
                      
                k = 1;
                  
                while((j + 1) != k)
                    fc[i][j] /= f[k++];
            }
        }
    }
      
    // Function to print Fibonomial Triangle.
    static void printFibonomialTriangle(int n)
    {
        int f[] = new int[N+1];
          
        // Finding the fibonacci series.
        fib(f, n);
          
        // to store triangle value.
        int dp[][] = new int[N + 1][N + 1];
          
        // initalising the 0th element of each row 
        // and diagonal element equal to 0.
        for (int i = 0; i <= n; i++)
            dp[i][0] = dp[i][i] = 1;
              
        // for each row.
        for (int i = 1; i <= n; i++)
        {
            // for each column.
            for (int j = 1; j < i; j++)
              
                // finding each element using recurrence 
                // relation.
                dp[i][j] = f[i - j + 1] * dp[i - 1][j - 1] +
                                           f[j-1]*dp[i-1][j]; 
        }
          
        // printing the Fibonomial Triangle.
        for (int i = 0; i <= n; i++)
        {
            for (int j = 0; j <= i; j++)     
                System.out.print(dp[i][j] + " ");     
            System.out.println();
        }
}
  
    // Driver code
    public static void main (String[] args)
    {
        int n = 6;
          
        printFibonomialTriangle(n);
    }
}
  
// This code is contributed by Anant Agarwal.

Python3

# Python3 Program to print Fibonomial 
# Triangle of height n.
N = 6;
  
# Function to produce Fibonacci Series.
def fib(f, n):
  
    # 0th and 1st number of the 
    # series are 0 and 1
    f[0] = 0;
    f[1] = 1;
  
    for i in range(2, n + 1):
      
        # Add the previous 2 numbers in 
        # the series and store it 
        f[i] = f[i - 1] + f[i - 2]; 
  
# Function to produce fibonomial
# coefficient
def fibcoef(fc, f, n):
  
    for i in range(n + 1):
        fc[i][0] = 1;
          
    for i in range(1, n + 1):
        for j in range(1, i + 1):
            k = j;
            while(k > 0):
                k -= 1;
                fc[i][j] *= f[k];
                  
            k = 1;
              
            while((j + 1) != k):
                fc[i][j] /= f[k];
                k += 1;
  
# Function to print Fibonomial Triangle.
def printFibonomialTriangle(n):
  
    f = [0] * (N + 1);
      
    # Finding the fibonacci series.
    fib(f, n);
      
    # to store triangle value.
    dp = [[0 for x in range(N + 1)] 
             for y in range(N + 1)];
      
    # initalising the 0th element of each
    # row and diagonal element equal to 0.
    for i in range(n + 1):
        dp[i][0] = 1;
        dp[i][i] = 1;
          
    # for each row.
    for i in range(1, n + 1):
        # for each column.
        for j in range(1, i):
          
            # finding each element using 
            # recurrence relation.
            dp[i][j] = (f[i - j + 1] * dp[i - 1][j - 1] + 
                        f[j - 1] * dp[i - 1][j]); 
      
    # printing the Fibonomial Triangle.
    for i in range(n + 1):
        for j in range(i + 1):     
            print(dp[i][j], end = " ");     
        print("");
      
# Driver Code
n = 6; 
printFibonomialTriangle(n);
  
# This code is contributed by mits

C#

// C# Program to print Fibonomial 
// Triangle of height n.
using System;
  
class GFG
{
    static int N = 6;
      
    // Function to produce Fibonacci Series.
    static void fib(int []f, int n)
    {
        int i;
      
        /* 0th and 1st number of
        the series are 0 and 1*/
        f[0] = 0;
        f[1] = 1;
      
        for (i = 2; i <= n; i++)
          
            /* Add the previous 2 numbers in 
            the series and store it */
            f[i] = f[i - 1] + f[i - 2]; 
    }
      
    // Function to produce fibonomial coefficient
    static void fibcoef(int [,]fc, int []f, int n)
    {
        for (int i = 0; i <= n; i++)
            fc[i,0] = 1;
              
        for (int i = 1; i <= n; i++)
        {
            for (int j = 1; j <= i; j++)
            {
                int k = j;
                  
                while(k > 0)
                {
                    k--;
                    fc[i, j] *= f[k];
                }
                      
                k = 1;
                  
                while((j + 1) != k)
                    fc[i, j] /= f[k++];
            }
        }
    }
      
    // Function to print Fibonomial Triangle.
    static void printFibonomialTriangle(int n)
    {
        int []f = new int[N + 1];
          
        // Finding the fibonacci series.
        fib(f, n);
          
        // to store triangle value.
        int [,]dp = new int[N + 1, N + 1];
          
        // initalising the 0th element of each row 
        // and diagonal element equal to 0.
        for (int i = 0; i <= n; i++)
            dp[i, 0] = dp[i, i] = 1;
              
        // for each row.
        for (int i = 1; i <= n; i++)
        {
            // for each column.
            for (int j = 1; j < i; j++)
              
                // finding each element using recurrence 
                // relation.
                dp[i,j] = f[i - j + 1] * dp[i - 1,j - 1] +
                                    f[j - 1] * dp[i - 1, j]; 
        }
          
        // printing the Fibonomial Triangle.
        for (int i = 0; i <= n; i++)
        {
            for (int j = 0; j <= i; j++) 
            Console.Write(dp[i,j] + " "); 
            Console.WriteLine();
        }
}
  
    // Driver code
    public static void Main ()
    {
        int n = 6;
          
        printFibonomialTriangle(n);
    }
}
  
// This code is contributed by Vt_m.

输出：

1 
1 1 
1 1 1 
1 2 2 1 
1 3 6 3 1 
1 5 15 15 5 1 
1 8 40 60 40 8 1