关于斐波那契数的有趣事实
原文:https://www . geesforgeks . org/interior-facts-Fibonacci-numbers/
我们知道斐波那契数，Fn =Fn-1+Fn-2。 前几个斐波那契数是 0、1、1、2、3、5、8、13、21、34、55、89、144、233、377、610、987、…。。 这里有一些关于斐波那契数的有趣事实:
1。斐波那契数列最后几个数字的模式: 前几个斐波那契数列的最后几个数字是:
0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, ...
最后一串数字以 60 的周期长度重复(关于这个结果的解释，请参考这个)。
C
// C program to demonstrate that sequence of last
// digits of Fibonacci numbers repeats after 60.
#include<stdio.h>
#define max 100
int main()
{
long long int arr[max];
arr[0] = 0;
arr[1] = 1;
int i = 0;
// storing Fibonacci numbers
for (int i = 2; i < max; i++)
arr[i] = arr[i-1] + arr[i-2];
// Traversing through store numbers
for (int i = 1; i < max - 1; i++)
{
// Since first two number are 0 and 1
// so, if any two consecutive number encounter 0 and 1
// at their unit place, then it clearly means that
// number is repeating/ since we just have to find
// the sum of previous two number
if ((arr[i] % 10 == 0) && (arr[i+1] % 10 == 1))
break;
}
printf("Sequence is repeating after index %d", i);
}
Java 语言(一种计算机语言，尤用于创建网站)
// Java program to demonstrate that sequence of last
// digits of Fibonacci numbers repeats after 60.
class GFG{
static int max=100;
public static void main(String[] args)
{
long[] arr=new long[max];
arr[0] = 0;
arr[1] = 1;
int i=0;
// storing Fibonacci numbers
for (i = 2; i < max; i++)
arr[i] = arr[i-1] + arr[i-2];
// Traversing through store numbers
for (i = 1; i < max - 1; i++)
{
// Since first two number are 0 and 1
// so, if any two consecutive number encounter 0 and 1
// at their unit place, then it clearly means that
// number is repeating/ since we just have to find
// the sum of previous two number
if ((arr[i] % 10 == 0) && (arr[i+1] % 10 == 1))
break;
}
System.out.println("Sequence is repeating after index "+i);
}
}
// This code is contributed by mits
Python 3
# Python3 program to demonstrate that sequence of last
# digits of Fibonacci numbers repeats after 60.
if __name__=='__main__':
max = 100
arr = [0 for i in range(max)]
arr[0] = 0
arr[1] = 1
# storing Fibonacci numbers
for i in range(2, max):
arr[i] = arr[i - 1] + arr[i - 2]
# Traversing through store numbers
for i in range(1, max - 1):
# Since first two number are 0 and 1
# so, if any two consecutive number encounter 0 and 1
# at their unit place, then it clearly means that
# number is repeating/ since we just have to find
# the sum of previous two number
if((arr[i] % 10 == 0) and (arr[i + 1] % 10 == 1)):
break
print("Sequence is repeating after index", i)
# This code is contributed by
# Sanjit_Prasad
C
// C# program to demonstrate that sequence of last
// digits of Fibonacci numbers repeats after 60.
class GFG{
static int max=100;
public static void Main()
{
long[] arr=new long[max];
arr[0] = 0;
arr[1] = 1;
int i=0;
// storing Fibonacci numbers
for (i = 2; i < max; i++)
arr[i] = arr[i-1] + arr[i-2];
// Traversing through store numbers
for (i = 1; i < max - 1; i++)
{
// Since first two number are 0 and 1
// so, if any two consecutive number encounter 0 and 1
// at their unit place, then it clearly means that
// number is repeating/ since we just have to find
// the sum of previous two number
if ((arr[i] % 10 == 0) && (arr[i+1] % 10 == 1))
break;
}
System.Console.WriteLine("Sequence is repeating after index "+i);
}
}
// This code is contributed by mits
服务器端编程语言（Professional Hypertext Preprocessor 的缩写）
<?php
// php program to demonstrate that
// sequence of last digits of
// Fibonacci numbers repeats after
// 60\. global $MAX=100
$arr[0] = 0;
$arr[1] = 1;
// storing Fibonacci numbers
for ($i = 2; $i < 100; $i++)
$arr[$i] = $arr[$i-1] +
$arr[$i-2];
// Traversing through store
// numbers
for ($i = 1; $i <100 - 1; $i++)
{
// Since first two number are
// 0 and 1 so, if any two
// consecutive number encounter
// 0 and 1 at their unit place,
// then it clearly means that
// number is repeating/ since
// we just have to find the
// sum of previous two number
if (($arr[$i] % 10 == 0) &&
($arr[$i+1] % 10 == 1))
break;
}
echo "Sequence is repeating after",
" index ", $i;
// This code is contributed by ajit
?>
java 描述语言
<script>
// Javascript program to demonstrate that
// sequence of last digits of Fibonacci
// numbers repeats after 60.
var max = 100;
var arr = Array(max).fill(0);
arr[0] = 0;
arr[1] = 1;
var i = 0;
// Storing Fibonacci numbers
for(i = 2; i < max; i++)
arr[i] = arr[i - 1] + arr[i - 2];
// Traversing through store numbers
for(i = 1; i < max - 1; i++)
{
// Since first two number are 0 and 1
// so, if any two consecutive number encounter 0 and 1
// at their unit place, then it clearly means that
// number is repeating since we just have to find
// the sum of previous two number
if ((arr[i] % 10 == 0) && (arr[i + 1] % 10 == 1))
break;
}
// Driver code
document.write("Sequence is repeating after index " + i);
// This code is contributed by gauravrajput1
</script>
输出:
Sequence is repeating after index 60
2。斐波那契数的因素:仔细观察，我们可以观察到以下几点:
- 每第三个斐波那契数是 2 的倍数
- 每 4 个斐波那契数是 3 的倍数
- 每 5 个斐波那契数是 5 的倍数
- 每 6 个斐波那契数是 8 的倍数
详见本。
C++
// C++ program to demonstrate divisibility of Fibonacci
// numbers.
#include<iostream>
using namespace std;
#define MAX 90
int main()
{
// indexes variable stores index of number that
// is divisible by 2, 3, 5 and 8
long long int arr[MAX], index1[MAX], index2[MAX];
long long int index3[MAX], index4[MAX];
// storing fibonacci numbers
arr[0] = 0;
arr[1] = 1;
for (int i = 2; i < MAX; i++)
arr[i] = arr[i-1] + arr[i-2];
// c1 keeps track of number of index of number
// divisible by 2 and others c2, c3 and c4 for
// 3, 5 and 8
int c1 = 0, c2 = 0, c3 = 0, c4 = 0;
// separating fibonacci number into their
// respective array
for (int i = 0; i < MAX; i++)
{
if (arr[i] % 2 == 0)
index1[c1++] = i;
if (arr[i] % 3 == 0)
index2[c2++] = i;
if (arr[i] % 5 == 0)
index3[c3++] = i;
if (arr[i] % 8 == 0)
index4[c4++] = i;
}
// printing index arrays
cout<<"Index of Fibonacci numbers divisible by"
" 2 are :\n";
for (int i = 0; i < c1; i++)
cout<<" "<< index1[i];
cout<<"\n";
cout<<"Index of Fibonacci number divisible by"
" 3 are :\n";
for (int i = 0; i < c2; i++)
cout<<" "<< index2[i];
cout<<"\n";
cout<<"Index of Fibonacci number divisible by"
" 5 are :\n";
for (int i = 0; i < c3; i++)
cout<<" "<< index3[i];
cout<<"\n";
cout<<"Index of Fibonacci number divisible by"
" 8 are :\n";
for (int i = 0; i < c4; i++)
cout<<" "<<index4[i];
cout<<"\n";
}
// This code is contributed by shivanisinghss2110
C
// C program to demonstrate divisibility of Fibonacci
// numbers.
#include<stdio.h>
#define MAX 90
int main()
{
// indexes variable stores index of number that
// is divisible by 2, 3, 5 and 8
long long int arr[MAX], index1[MAX], index2[MAX];
long long int index3[MAX], index4[MAX];
// storing fibonacci numbers
arr[0] = 0;
arr[1] = 1;
for (int i = 2; i < MAX; i++)
arr[i] = arr[i-1] + arr[i-2];
// c1 keeps track of number of index of number
// divisible by 2 and others c2, c3 and c4 for
// 3, 5 and 8
int c1 = 0, c2 = 0, c3 = 0, c4 = 0;
// separating fibonacci number into their
// respective array
for (int i = 0; i < MAX; i++)
{
if (arr[i] % 2 == 0)
index1[c1++] = i;
if (arr[i] % 3 == 0)
index2[c2++] = i;
if (arr[i] % 5 == 0)
index3[c3++] = i;
if (arr[i] % 8 == 0)
index4[c4++] = i;
}
// printing index arrays
printf("Index of Fibonacci numbers divisible by"
" 2 are :\n");
for (int i = 0; i < c1; i++)
printf("%d ", index1[i]);
printf("\n");
printf("Index of Fibonacci number divisible by"
" 3 are :\n");
for (int i = 0; i < c2; i++)
printf("%d ", index2[i]);
printf("\n");
printf("Index of Fibonacci number divisible by"
" 5 are :\n");
for (int i = 0; i < c3; i++)
printf("%d ", index3[i]);
printf("\n");
printf("Index of Fibonacci number divisible by"
" 8 are :\n");
for (int i = 0; i < c4; i++)
printf("%d ", index4[i]);
printf("\n");
}
Java 语言(一种计算机语言，尤用于创建网站)
// Java program to demonstrate divisibility of Fibonacci
// numbers.
class GFG
{
static int MAX=90;
// Driver code
public static void main(String[] args)
{
// indexes variable stores index of number that
// is divisible by 2, 3, 5 and 8
long[] arr=new long[MAX];
long[] index1=new long[MAX];
long[] index2=new long[MAX];
long[] index3=new long[MAX];
long[] index4=new long[MAX];
// storing fibonacci numbers
arr[0] = 0;
arr[1] = 1;
for (int i = 2; i < MAX; i++)
arr[i] = arr[i - 1] + arr[i - 2];
// c1 keeps track of number of index of number
// divisible by 2 and others c2, c3 and c4 for
// 3, 5 and 8
int c1 = 0, c2 = 0, c3 = 0, c4 = 0;
// separating fibonacci number into their
// respective array
for (int i = 0; i < MAX; i++)
{
if (arr[i] % 2 == 0)
index1[c1++] = i;
if (arr[i] % 3 == 0)
index2[c2++] = i;
if (arr[i] % 5 == 0)
index3[c3++] = i;
if (arr[i] % 8 == 0)
index4[c4++] = i;
}
// printing index arrays
System.out.print("Index of Fibonacci numbers divisible by" +
" 2 are :\n");
for (int i = 0; i < c1; i++)
System.out.print(index1[i] + " ");
System.out.print("\n");
System.out.print("Index of Fibonacci number divisible by" +
" 3 are :\n");
for (int i = 0; i < c2; i++)
System.out.print(index2[i] + " ");
System.out.print("\n");
System.out.print("Index of Fibonacci number divisible by" +
" 5 are :\n");
for (int i = 0; i < c3; i++)
System.out.print(index3[i] + " ");
System.out.print("\n");
System.out.print("Index of Fibonacci number divisible by" +
" 8 are :\n");
for (int i = 0; i < c4; i++)
System.out.print(index4[i] + " ");
System.out.print("\n");
}
}
// This code is contributed by mits
Python 3
# Python3 program to demonstrate divisibility
# of Fibonacci numbers.
MAX = 90;
# indexes variable stores index of number
# that is divisible by 2, 3, 5 and 8
arr = [0] * (MAX);
index1 = [0] * (MAX);
index2 = [0] * (MAX);
index3 = [0] * (MAX);
index4 = [0] * (MAX);
# storing fibonacci numbers
arr[0] = 0;
arr[1] = 1;
for i in range(2, MAX):
arr[i] = arr[i - 1] + arr[i - 2];
# c1 keeps track of number of index
# of number divisible by 2 and others
# c2, c3 and c4 for 3, 5 and 8
c1, c2, c3, c4 = 0, 0, 0, 0;
# separating fibonacci number into
# their respective array
for i in range(MAX):
if (arr[i] % 2 == 0):
index1[c1] = i;
c1 += 1;
if (arr[i] % 3 == 0):
index2[c2] = i;
c2 += 1;
if (arr[i] % 5 == 0):
index3[c3] = i;
c3 += 1;
if (arr[i] % 8 == 0):
index4[c4] = i;
c4 += 1;
# printing index arrays
print("Index of Fibonacci numbers",
"divisible by 2 are :");
for i in range(c1):
print(index1[i], end = " ");
print("");
print("Index of Fibonacci number",
"divisible by 3 are :");
for i in range(c2):
print(index2[i], end = " ");
print("");
print("Index of Fibonacci number",
"divisible by 5 are :");
for i in range(c3):
print(index3[i], end = " ");
print("");
print("Index of Fibonacci number",
"divisible by 8 are :");
for i in range(c4):
print(index4[i], end = " ");
print("");
# This code is contributed by mits
C
// C# program to demonstrate divisibility
// of Fibonacci numbers.
class GFG{
static int MAX = 90;
static void Main()
{
// indexes variable stores index of number that
// is divisible by 2, 3, 5 and 8
long[] arr = new long[MAX];
long[] index1 = new long[MAX];
long[] index2 = new long[MAX];
long[] index3 = new long[MAX];
long[] index4 = new long[MAX];
// storing fibonacci numbers
arr[0] = 0;
arr[1] = 1;
for (int i = 2; i < MAX; i++)
arr[i] = arr[i-1] + arr[i-2];
// c1 keeps track of number of index of number
// divisible by 2 and others c2, c3 and c4 for
// 3, 5 and 8
int c1 = 0, c2 = 0, c3 = 0, c4 = 0;
// separating fibonacci number into their
// respective array
for (int i = 0; i < MAX; i++)
{
if (arr[i] % 2 == 0)
index1[c1++] = i;
if (arr[i] % 3 == 0)
index2[c2++] = i;
if (arr[i] % 5 == 0)
index3[c3++] = i;
if (arr[i] % 8 == 0)
index4[c4++] = i;
}
// printing index arrays
System.Console.Write("Index of Fibonacci numbers" +
"divisible by 2 are :\n");
for (int i = 0; i < c1; i++)
System.Console.Write(index1[i]+" ");
System.Console.Write("\n");
System.Console.Write("Index of Fibonacci number "+
" divisible by 3 are :\n");
for (int i = 0; i < c2; i++)
System.Console.Write(index2[i]+" ");
System.Console.Write("\n");
System.Console.Write("Index of Fibonacci number "+
"divisible by 5 are :\n");
for (int i = 0; i < c3; i++)
System.Console.Write(index3[i]+" ");
System.Console.Write("\n");
System.Console.Write("Index of Fibonacci number "+
"divisible by 8 are :\n");
for (int i = 0; i < c4; i++)
System.Console.Write(index4[i]+" ");
System.Console.Write("\n");
}
}
// This code is contributed by mits
服务器端编程语言（Professional Hypertext Preprocessor 的缩写）
<?php
// PHP program to demonstrate divisibility
// of Fibonacci numbers.
$MAX = 90;
// indexes variable stores index of number
// that is divisible by 2, 3, 5 and 8
$arr = array($MAX);
$index1 = array($MAX);
$index2 = array($MAX);
$index3 = array($MAX);
$index4 = array($MAX);
// storing fibonacci numbers
$arr[0] = 0;
$arr[1] = 1;
for ($i = 2; $i < $MAX; $i++)
{
$arr[$i] = $arr[$i - 1] + $arr[$i - 2];
}
// c1 keeps track of number of index of
// number divisible by 2 and others
// c2, c3 and c4 for 3, 5 and 8
$c1 = 0;
$c2 = 0;
$c3 = 0;
$c4 = 0;
// separating fibonacci number into
// their respective array
for ($i = 0; $i < $MAX; $i++)
{
if ($arr[$i] % 2 == 0)
$index1[$c1++] = $i;
if ($arr[$i] % 3 == 0)
$index2[$c2++] = $i;
if ($arr[$i] % 5 == 0)
$index3[$c3++] = $i;
if ($arr[$i] % 8 == 0)
$index4[$c4++] = $i;
}
// printing index arrays
echo "Index of Fibonacci numbers divisible by" .
" 2 are :\n";
for ($i = 0; $i < $c1; $i++)
echo $index1[$i] . " ";
echo "\n";
echo "Index of Fibonacci number divisible by" .
" 3 are :\n";
for ($i = 0; $i < $c2; $i++)
echo $index2[$i] . " ";
echo "\n";
echo "Index of Fibonacci number divisible by" .
" 5 are :\n";
for ($i = 0; $i < $c3; $i++)
echo $index3[$i] . " ";
echo "\n";
echo "Index of Fibonacci number divisible by" .
" 8 are :\n";
for ($i = 0; $i < $c4; $i++)
echo $index4[$i] . " ";
echo "\n";
// This code is contributed by mits
?>
java 描述语言
// JavaScript program to demonstrate divisibility of Fibonacci
// numbers.
var MAX=90;
// Driver code
// indexes variable stores index of number that
// is divisible by 2, 3, 5 and 8
var arr=new Array(MAX);
var index1= new Array(MAX);
var index2= new Array(MAX);
var index3= new Array(MAX);
var index4= new Array(MAX);
// storing fibonacci numbers
arr[0] = 0;
arr[1] = 1;
for (var i = 2; i < MAX; i++)
arr[i] = arr[i - 1] + arr[i - 2];
// c1 keeps track of number of index of number
// divisible by 2 and others c2, c3 and c4 for
// 3, 5 and 8
var c1 = 0, c2 = 0, c3 = 0, c4 = 0;
// separating fibonacci number into their
// respective array
for (var i = 0; i < MAX; i++)
{
if (arr[i] % 2 == 0)
index1[c1++] = i;
if (arr[i] % 3 == 0)
index2[c2++] = i;
if (arr[i] % 5 == 0)
index3[c3++] = i;
if (arr[i] % 8 == 0)
index4[c4++] = i;
}
// printing index arrays
document.write("Index of Fibonacci numbers divisible by" +
" 2 are :\n");
for (var i = 0; i < c1; i++)
document.write(index1[i] + " ");
document.write("\n");
document.write("Index of Fibonacci number divisible by" +
" 3 are :\n");
for (var i = 0; i < c2; i++)
document.write(index2[i] + " ");
document.write("\n");
document.write("Index of Fibonacci number divisible by" +
" 5 are :\n");
for (var i = 0; i < c3; i++)
document.write(index3[i] + " ");
document.write("\n");
document.write("Index of Fibonacci number divisible by" +
" 8 are :\n");
for (var i = 0; i < c4; i++)
document.write(index4[i] + " ");
document.write("\n");
// This code is contributed by shivanisinghss2110
输出:
Index of Fibonacci numbers divisible by 2 are :
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45
48 51 54 57 60 63 66 69 72 75 78 81 84 87
Index of Fibonacci number divisible by 3 are :
0 4 8 12 16 20 24 28 32 36 40 44 48 52
56 60 64 68 72 76 80 84 88
Index of Fibonacci number divisible by 5 are :
0 5 10 15 20 25 30 35 40 45 50
55 60 65 70 75 80 85
Index of Fibonacci number divisible by 8 are :
0 6 12 18 24 30 36 42 48
54 60 66 72 78 84
3。带指数因子的斐波那契数:我们有一些斐波那契数像 F(1) = 1 可被 1 整除，F(5) = 5 可被 5 整除，F(12) = 144 可被 12 整除，F(24) = 46368 可被 24 整除，F(25) = 75025 可被 25 整除。这种类型的索引号遵循一定的模式。首先，让我们看看这些指数: 1，5，12，24，25，36，48，60，72，84，96，108，120，125，132，…..
根据观察，这个级数是由 12 的倍数以及满足幂(5，k)条件的所有数组成的，其中 k = 0，1，2，3，4，5，6，7，…。
C++
// C++ program to demonstrate that Fibonacci numbers
// that are divisible by their indexes have indexes
// as either power of 5 or multiple of 12.
#include<iostream>
using namespace std;
#define MAX 100
int main()
{
// storing Fibonacci numbers
long long int arr[MAX];
arr[0] = 0;
arr[1] = 1;
for (int i = 2; i < MAX; i++)
arr[i] = arr[i-1] + arr[i-2];
cout<<"Fibonacci numbers divisible by "
"their indexes are :\n";
for (int i = 1; i < MAX; i++)
if (arr[i] % i == 0)
cout<<" "<< i;
}
// This code is contributed by shivanisinghss2110
C
// C program to demonstrate that Fibonacci numbers
// that are divisible by their indexes have indexes
// as either power of 5 or multiple of 12.
#include<stdio.h>
#define MAX 100
int main()
{
// storing Fibonacci numbers
long long int arr[MAX];
arr[0] = 0;
arr[1] = 1;
for (int i = 2; i < MAX; i++)
arr[i] = arr[i-1] + arr[i-2];
printf("Fibonacci numbers divisible by "
"their indexes are :\n");
for (int i = 1; i < MAX; i++)
if (arr[i] % i == 0)
printf("%d ", i);
}
Java 语言(一种计算机语言，尤用于创建网站)
// Java program to demonstrate that Fibonacci numbers
// that are divisible by their indexes have indexes
// as either power of 5 or multiple of 12.
class GFG
{
static int MAX = 100;
public static void main(String[] args)
{
// storing Fibonacci numbers
long[] arr = new long[MAX];
arr[0] = 0;
arr[1] = 1;
for (int i = 2; i < MAX; i++)
arr[i] = arr[i - 1] + arr[i - 2];
System.out.print("Fibonacci numbers divisible by "+
"their indexes are :\n");
for (int i = 1; i < MAX; i++)
if (arr[i] % i == 0)
System.out.print(i + " ");
}
}
// This code is contributed by mits
Python 3
# Python3 program to demonstrate that Fibonacci numbers
# that are divisible by their indexes have indexes
# as either power of 5 or multiple of 12.
if __name__=='__main__':
MAX = 100
# storing Fibonacci numbers
arr = [0 for i in range(MAX)]
arr[0] = 0
arr[1] = 1
for i in range(2, MAX):
arr[i] = arr[i - 1] + arr[i - 2]
print("Fibonacci numbers divisible by their indexes are :")
for i in range(1, MAX):
if(arr[i] % i == 0):
print(i,end=" ")
# This code is contributed by
# Sanjit_Prasad
C
// C# program to demonstrate that Fibonacci
// numbers that are divisible by their
// indexes have indexes as either power of 5
// or multiple of 12.
using System;
class GFG
{
static int MAX = 100;
static void Main()
{
// storing Fibonacci numbers
long[] arr = new long[MAX];
arr[0] = 0;
arr[1] = 1;
for (int i = 2; i < MAX; i++)
arr[i] = arr[i - 1] + arr[i - 2];
Console.Write("Fibonacci numbers divisible by " +
"their indexes are :\n");
for (int i = 1; i < MAX; i++)
if (arr[i] % i == 0)
System.Console.Write(i+" ");
}
}
// This code is contributed by mits
java 描述语言
// JavaScript program to demonstrate that Fibonacci numbers
// that are divisible by their indexes have indexes
// as either power of 5 or multiple of 12.
var MAX = 100;
// storing Fibonacci numbers
var arr = new Array(MAX);
arr[0] = 0;
arr[1] = 1;
for (var i = 2; i < MAX; i++)
arr[i] = arr[i - 1] + arr[i - 2];
document.write("Fibonacci numbers divisible by their indexes are :");
for (var i = 1; i < MAX; i++)
if (arr[i] % i == 0)
document.write(i + " ");
// This code is contributed by shivanisinghss2110
输出:
Fibonacci numbers divisible by their indexes are :
1 5 12 24 25 36 48 60 72 96
4。f(n-1) f(n+1)–f(n) f(n)的值为(-1) n 。详见卡西尼身份。
参考: http://www . mathematics . surrey . AC . uk/hosted-sites/R .克诺特/Fibonacci/fib mathematics . html 本文由 Aditya Kumar 供稿。如果你喜欢 GeeksforGeeks 并想投稿，你也可以用contribute.geeksforgeeks.org写一篇文章或者把你的文章邮寄到 review-team@geeksforgeeks.org。看到你的文章出现在极客博客主页上，帮助其他极客。 如果发现有不正确的地方，或者想分享更多关于上述话题的信息，请写评论。
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