Interesting facts about Fibonacci numbers

3.4

We know Fibonacci number, Fn = Fn-1 + Fn-2.

First few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …. .

Here are some interesting facts about Fibonacci number :

1. Pattern in Last digits of Fibonacci numbers :
Last digits of first few Fibonacci Numbers are :

0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, ... 

The series of last digits repeats with a cycle length of 60 (Refer this for explanations of this result).

// 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;

    // 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);
}

Output:

Sequence is repeating after index 60

2. Factors of Fibonacci number : On careful observation, we can observe the following thing :

  • Every 3-rd Fibonacci number is a multiple of 2
  • Every 4-th Fibonacci number is a multiple of 3
  • Every 5-th Fibonacci number is a multiple of 5
  • Every 6-th Fibonacci number is a multiple of 8

Refer this for details.

// C program to demonstrate divisibility of Fibonacci
// numbers.
#include<stdio.h>
#define MAX 100

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");
}

Output:

Index of Fibonacci number 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  90  93  96  99
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  92  96  
98
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  90
Index of Fibonacci number divisible by 8 are :
0  6  12  18  24  30  36  42  48  54  60  66  
72  78  84  90  96

3. Fibonacci number with index number factor : We have some Fibonacci number like F(1) = 1 which is divisible by 1, F(5) = 5 which is divisible by 5, F(12) = 144 which is divisible by 12, F(24) = 46368 which is divisible by 24, F(25) = 75025 which is divisible by 25. This type of index number follow a certain pattern. First, let’s keep a look on those index number :
1, 5, 12, 24, 25, 36, 48, 60, 72, 84, 96, 108, 120, 125, 132, …..
On observing it, this series is made up of every number that is multiple of 12 as well as all the number that satisfies the condition of pow(5, k), where k = 0, 1, 2, 3, 4, 5, 6, 7, …….

// 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);
}

Output:

Fibonacci numbers divisible by their indexes are :
1  5  12  24  25  36  48  60  72  96

4. Value of f(n-1)*f(n+1) – f(n)*f(n) is (-1)n. Please refer Cassini’s Identity for details.

Reference :
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html

This article is contributed by Aditya Kumar. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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