Open In App
Related Articles

Find Index of given fibonacci number in constant time

Improve
Improve
Improve
Like Article
Like
Save Article
Save
Report issue
Report

We are given a Fibonacci number. The first few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ….. 
We have to find the index of the given Fibonacci number, i.e. like Fibonacci number 8 is at index 6. 

Examples : 

Input : 13
Output : 7
Input : 34
Output : 9

Method 1 (Simple) : A simple approach is to find Fibonacci numbers up to the given Fibonacci numbers and count the number of iterations performed.

C++

// A simple C++ program to find index of given
// Fibonacci number.
#include<bits/stdc++.h>
 
int findIndex(int n)
{
    // if Fibonacci number is less than 2,
    // its index will be same as number
    if (n <= 1)
        return n;
 
    int a = 0, b = 1, c = 1;
    int res = 1;
 
    // iterate until generated fibonacci number
    // is less than given fibonacci number
    while (c < n)
    {
        c = a + b;
         
        // res keeps track of number of generated
        // fibonacci number
 
        res++;
        a = b;
        b = c;
    }
    return res;
}
 
// Driver program to test above function
int main()
{
    int result = findIndex(21);
    printf("%d\n", result);
}
 
// This code is contributed by Saket Kumar

                    

Java

// A simple Java program to find index of
// given Fibonacci number.
import java.io.*;
 
class GFG {
     
    static int findIndex(int n)
    {
         
        // if Fibonacci number is less
        // than 2, its index will be
        // same as number
        if (n <= 1)
            return n;
     
        int a = 0, b = 1, c = 1;
        int res = 1;
     
        // iterate until generated fibonacci
        // number is less than given
        // fibonacci number
        while (c < n)
        {
            c = a + b;
             
            // res keeps track of number of
            // generated fibonacci number
            res++;
            a = b;
            b = c;
        }
         
        return res;
    }
     
    // Driver program to test above function
    public static void main (String[] args)
    {
        int result = findIndex(21);
        System.out.println( result);
    }
}
 
// This code is contributed by anuj_67.

                    

Python3

# A simple Python 3 program to find
# index of given Fibonacci number.
 
def findIndex(n) :
     
    # if Fibonacci number is less than 2,
    # its index will be same as number
    if (n <= 1) :
        return n
  
    a = 0
    b = 1
    c = 1
    res = 1
  
    # iterate until generated fibonacci number
    # is less than given fibonacci number
    while (c < n) :
        c = a + b
          
        # res keeps track of number of 
        # generated fibonacci number
        res = res + 1
        a = b
        b = c
         
    return res
 
# Driver program to test above function
result = findIndex(21)
print(result)
 
# this code is contributed by Nikita Tiwari

                    

C#

// A simple C# program to
// find index of given
// Fibonacci number.
using System;
class GFG
{
    static int findIndex(int n)
    {
         
        // if Fibonacci number
        // is less than 2, its
        // index will be same
        // as number
        if (n <= 1)
            return n;
     
        int a = 0, b = 1, c = 1;
        int res = 1;
     
        // iterate until generated
        // fibonacci number is less
        // than given fibonacci number
        while (c < n)
        {
            c = a + b;
             
            // res keeps track of
            // number of generated
            // fibonacci number
            res++;
            a = b;
            b = c;
        }
         
        return res;
    }
     
    // Driver Code
    public static void Main ()
    {
        int result = findIndex(21);
        Console.WriteLine(result);
    }
}
 
// This code is contributed
// by anuj_67.

                    

Javascript

<script>
 
// A simple Javascript program to
// find index of given
// Fibonacci number.
function findIndex(n)
{
       
    // If Fibonacci number
    // is less than 2, its
    // index will be same
    // as number
    if (n <= 1)
        return n;
   
    let a = 0, b = 1, c = 1;
    let res = 1;
   
    // Iterate until generated
    // fibonacci number is less
    // than given fibonacci number
    while (c < n)
    {
        c = a + b;
           
        // res keeps track of
        // number of generated
        // fibonacci number
        res++;
        a = b;
        b = c;
    }
    return res;
}
 
// Driver code
let result = findIndex(21);
document.write(result);
 
// This code is contributed by decode2207
 
</script>

                    

PHP

<?php
// A simple PHP program to
// find index of given
// Fibonacci number.
 
function findIndex($n)
{
     
    // if Fibonacci number
    // is less than 2,
    // its index will be
    // same as number
    if ($n <= 1)
        return $n;
 
    $a = 0; $b = 1; $c = 1;
    $res = 1;
 
    // iterate until generated
    // fibonacci number
    // is less than given
    // fibonacci number
    while ($c < $n)
    {
        $c = $a + $b;
         
        // res keeps track of
        // number of generated
        // fibonacci number
 
        $res++;
        $a = $b;
        $b = $c;
    }
    return $res;
}
 
// Driver Code
$result = findIndex(21);
echo($result);
 
// This code is contributed by Ajit.
?>

                    

Output
8





Method 2 (Formula based)
But here, we needed to generate all the Fibonacci numbers up to a provided Fibonacci number. But, there is a quick solution to this problem. Let’s see how! Note that computing the log of a number is an O(1) operation on most platforms.


The Fibonacci number is described as, 
Fn = 1 / sqrt(5) (pow(a,n) – pow(b,n)) where 
a = 1 / 2 ( 1 + sqrt(5) ) and b = 1 / 2 ( 1 – sqrt(5) )

On neglecting pow(b, n) which is very small because of the large value of n, we get 
n = round { 2.078087 * log(Fn) + 1.672276 } 
where round means round to the nearest integer.

Below is the implementation of the above idea. 

C++

// C++ program to find index of given Fibonacci
// number
#include<bits/stdc++.h>
 
int findIndex(int n)
{
    float fibo = 2.078087 * log(n) + 1.672276;
 
    // returning rounded off value of index
    return round(fibo);
}
 
// Driver program to test above function
int main()
{
    int n = 55;
    printf("%d\n", findIndex(n));
}

                    

Java

// A simple Java program to find index of given
// Fibonacci number
public class Fibonacci
{
 
  static int findIndex(int n)
  {
    float fibo = 2.078087F * (float) Math.log(n) + 1.672276F;
 
    // returning rounded off value of index
    return Math.round(fibo);
  }
 
  public static void main(String[] args)
  {
    int result = findIndex(55);
    System.out.println(result);
  }
}

                    

Python3

# Python 3 program to find index of given Fibonacci
# number
 
import math
def findIndex(n) :
    fibo = 2.078087 * math.log(n) + 1.672276
  
    # returning rounded off value of index
    return round(fibo)
 
 
# Driver program to test above function
n = 21
print(findIndex(n))
 
 
# This code is contributed by Nikita Tiwari.

                    

C#

// A simple C# program to find
// index of given Fibonacci number
using System;
 
class Fibonacci {
 
static int findIndex(int n)
{
    float fibo = 2.078087F * (float) Math.Log(n) +
                                        1.672276F;
 
    // returning rounded off value of index
    return (int)(Math.Round(fibo));
}
 
  // Driver code
  public static void Main()
  {
    int result = findIndex(55);
    Console.Write(result);
  }
}
 
// This code is contributed by nitin mittal

                    

Javascript

<script>
 
// A simple Javascript program to find
// index of given Fibonacci number
 
function findIndex(n)
{
    var fibo = 2.078087 * parseFloat(Math.log(n)) + 1.672276;
     
    // Returning rounded off value of index
    return Math.round(fibo);
}
 
// Driver code
var result = findIndex(55);
document.write(result);
 
// This code is contributed by Ankita saini
 
</script>

                    

PHP

<?php
// PHP program to find index
// of given Fibonacci Number
 
function findIndex($n)
{
    $fibo = 2.078087 * log($n) + 1.672276;
 
    // returning rounded off
    // value of index
    return round($fibo);
}
 
// Driver code
$n = 55;
echo(findIndex($n));
 
// This code is contributed by Ajit.
?>

                    

Output
10





Time complexity: O(1)
Auxiliary space: O(1)

Approach:

we can solve this problem using the formula for the nth Fibonacci number, which is:

F(n) = (pow((1+sqrt(5))/2, n) – pow((1-sqrt(5))/2, n)) / sqrt(5)

We can derive the index of a given Fibonacci number using this formula. We can iterate over the values of n and calculate the corresponding Fibonacci number using the above formula until we find a Fibonacci number that is greater than or equal to the given number. At this point, we can return the index of the Fibonacci number that matches the given number.

Below is the implementation of the above approach:

C++

#include <iostream>
#include <cmath>
using namespace std;
 
int findIndex(int n) {
    double phi = (1 + sqrt(5)) / 2;
    int index = round(log(n * sqrt(5) + 0.5) / log(phi));
    int fib = round((pow(phi, index) - pow(1 - phi, index)) / sqrt(5));
    if (fib == n)
        return index;
    else
        return -1; // n is not a Fibonacci number
}
 
int main() {
    int n = 34;
    int index = findIndex(n);
    cout << "The index of " << n << " is " << index << endl;
    return 0;
}

                    

Java

//Java code for the above approach
import java.util.*;
 
public class FibonacciIndex {
 
    public static int findIndex(int n) {
        double phi = (1 + Math.sqrt(5)) / 2;
        int index = (int) Math.round(Math.log(n * Math.sqrt(5) + 0.5) / Math.log(phi));
        int fib = (int) Math.round((Math.pow(phi, index) - Math.pow(1 - phi, index)) / Math.sqrt(5));
        if (fib == n)
            return index;
        else
            return -1; // n is not a Fibonacci number
    }
 
    public static void main(String[] args) {
        int n = 34;
        int index = findIndex(n);
        System.out.println("The index of " + n + " is " + index);
    }
}

                    

Python3

import math
 
def find_index(n):
    phi = (1 + math.sqrt(5)) / 2
    index = round(math.log(n * math.sqrt(5) + 0.5) / math.log(phi))
    fib = round((math.pow(phi, index) - math.pow(1 - phi, index)) / math.sqrt(5))
    if fib == n:
        return index
    else:
        return -1  # n is not a Fibonacci number
 
def main():
    n = 34
    index = find_index(n)
    print(f"The index of {n} is {index}")
 
if __name__ == "__main__":
    main()

                    

C#

using System;
 
class Program
{
    // Function to find the index of a number in the Fibonacci sequence
    static int FindIndex(int n)
    {
        double phi = (1 + Math.Sqrt(5)) / 2; // Golden ratio
         
        // Calculate the index using the formula for Fibonacci numbers
        int index = (int)Math.Round(Math.Log(n * Math.Sqrt(5) + 0.5) / Math.Log(phi));
         
        // Calculate the Fibonacci number at the found index
        int fib = (int)Math.Round((Math.Pow(phi, index) - Math.Pow(1 - phi, index)) / Math.Sqrt(5));
         
        // Check if the calculated Fibonacci number is equal to n
        if (fib == n)
            return index;
        else
            return -1; // n is not a Fibonacci number
    }
 
    static void Main()
    {
        int n = 34;
        int index = FindIndex(n);
        Console.WriteLine("The index of " + n + " is " + index);
    }
}

                    

Javascript

// Function to find the index of a number in the Fibonacci sequence
function findIndex(n) {
    const phi = (1 + Math.sqrt(5)) / 2;
    const index = Math.round(Math.log(n * Math.sqrt(5) + 0.5) / Math.log(phi));
    const fib = Math.round((Math.pow(phi, index) - Math.pow(1 - phi, index)) / Math.sqrt(5));
 
    if (fib === n) {
        return index;
    } else {
        return -1; // n is not a Fibonacci number
    }
}
 
// Main function to test the findIndex function
function main() {
    const n = 34;
    const index = findIndex(n);
    console.log("The index of " + n + " is " + index);
}
 
main();

                    

Output
The index of 34 is 9






Time Complexity: O(1) as it involves only a few arithmetic operations.
Space Complexity: O(1) as it uses only a constant amount of memory to store the variables.

This article is contributed by Aditya Kumar.

 



Last Updated : 26 Oct, 2023
Like Article
Save Article
Previous
Next
Share your thoughts in the comments
Similar Reads