Find F(n) when F(i) and F(j) of a sequence are given


Given five integers i, Fi, j, Fj and N. Where Fi and Fj are the ith and jth term of a sequence which follows the Fibonacci recurrence i.e. FN = FN – 1 + FN – 2. The task is to find the Nth term of the original sequence.

Examples:

Input: i = 3, F3 = 5, j = -1, F-1 = 4, N = 5
Output: 12
Fibonacci sequence can be reconstructed using known values:
…, F-1 = 4, F0 = -1, F1 = 3, F2 = 2, F3 = 5, F4 = 7, F5 = 12, …

Input: i = 0, F0 = 1, j = 1, F1 = 4, N = -2
Output: -2



Approach: Note that, if the two consecutive terms of the Fibonacci sequence are known then the Nth term can easily be determined. Assuming i < j, as per Fibonacci condition:

  Fi+1 = 1*Fi+1 + 0*Fi
  Fi+2 = 1*Fi+1 + 1*Fi
  Fi+3 = Fi+2 + Fi+1 = 2*Fi+1 + 1*Fi
  Fi+4 = Fi+3 + Fi+2 = 3*Fi+1 + 2*Fi
  Fi+5 = Fi+4 + Fi+3 = 5*Fi+1 + 3*Fi
  .. .. ..
and so on

Note that, the coefficients of Fi+1 and Fi in the above set of equations are nothing but the terms of Standard Fibonacci Sequence.

So, considering the Standard Fibonacci sequence i.e. f0 = 0, f1 = 1, f2 = 1, f3 = 2, f4 = 3, … ; we can generalize, the above set of equations (for k > 0) as:

  Fi+k = fk*Fi+1 + fk-1*Fi   …(1)

Now consider,

  k = j-i   …(2)

Now, substituting eq.(2) in eq.(1), we get:

  Fj = fj-i*Fi+1 + fj-i-1*Fi

Hence, we can calculate Fi+1 from known values of Fi and Fj. Now that we know two consecutive terms of sequence F, we can easily reconstruct F and determine the value of FN.

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include<bits/stdc++.h>
  
using namespace std;
  
// Function to calculate kth Fibonacci number
// in the standard Fibonacci sequence
int fibonacci(int k)
{
    int a = 0, b = 1, c = 0;
    if( k == 0)
        return a;
    if (k == 1)
        return b;
    for(int i = 2; i < k + 1; i++)
    {
        c = a + b;
        a = b;
        b = c;
    }
    return c;
}
  
// Function to determine the value of F(n)
int determineFn(int i, int Fi, int j, int Fj, int n)
{
    if (j < i)
    {
        swap(i, j);
        swap(Fi, Fj);
    }
  
    // Find the value of F(i + 1)
    // from F(i) and F(j)
    int Fi1 = (Fj - fibonacci(j - i - 1) * Fi) / 
                    fibonacci(j - i);
  
    // When n is equal to i
    int Fn = 0;
    if (n == i)
        Fn = Fi;
  
    // When n is greater than i
    else if (n > i) 
    {
        int b = Fi;
        Fn = Fi1;
        n = n - 1;
        while (n != i)
        {
            n = n - 1;
            int a = b;
            b = Fn;
            Fn = a + b;
        }
    }
      
    // When n is smaller than i
    else 
    {
        int a = Fi;
        int b = Fi1;
        while (n != i)
        {
            n = n + 1;
            Fn = b - a;
            b = a;
            a = Fn;
        }
    }
    return Fn;
}
  
// Driver code
int main()
{
    int i = 3;
    int Fi = 5;
    int j = -1;
    int Fj = 4;
    int n = 5;
    cout << (determineFn(i, Fi, j, Fj, n));
}
  
// This code is contributed by Mohit Kumar

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Java

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// Java implementation of the approach
import java.util.*;
  
class GFG
{
  
    // Function to calculate kth Fibonacci number
    // in the standard Fibonacci sequence
    static int fibonacci(int k)
    {
        int a = 0, b = 1, c = 0;
        if (k == 0)
            return a;
        if (k == 1)
            return b;
        for (int i = 2; i < k + 1; i++)
        {
            c = a + b;
            a = b;
            b = c;
        }
        return c;
    }
  
    // Function to determine the value of F(n)
    static int determineFn(int i, int Fi,       
                           int j, int Fj, int n) 
    {
        if (j < i) 
        {
  
            // Swap i, j
            i = i + j;
            j = i - j;
            i = i - j;
  
            // swap Fi, Fj
            Fi = Fi + Fj;
            Fj = Fi - Fj;
            Fi = Fi - Fj;
        }
  
        // Find the value of F(i + 1)
        // from F(i) and F(j)
        int Fi1 = (Fj - fibonacci(j - i - 1) * Fi) / 
                        fibonacci(j - i);
  
        // When n is equal to i
        int Fn = 0;
        if (n == i)
            Fn = Fi;
  
        // When n is greater than i
        else if (n > i)
        {
            int b = Fi;
            Fn = Fi1;
            n = n - 1;
            while (n != i)
            {
                n = n - 1;
                int a = b;
                b = Fn;
                Fn = a + b;
            }
        }
  
        // When n is smaller than i
        else
        {
            int a = Fi;
            int b = Fi1;
            while (n != i) 
            {
                n = n + 1;
                Fn = b - a;
                b = a;
                a = Fn;
            }
        }
        return Fn;
    }
  
    // Driver Code
    public static void main(String[] args) 
    {
        int i = 3;
        int Fi = 5;
        int j = -1;
        int Fj = 4;
        int n = 5;
        System.out.println(determineFn(i, Fi, j, Fj, n));
    }
}
  
// This code is contributed by
// sanjeev2552

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Python3

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# Python3 implementation of the approach
  
# Function to calculate kth Fibonacci number 
# in the standard Fibonacci sequence
def fibonacci(k):
    a = 0
    b = 1
    if k == 0:
        return a
    if k == 1:
        return b
    for i in range(2, k + 1):
        c = a + b
        a = b
        b = c
    return c
  
# Function to determine
# the value of F(n)
def determineFn(i, Fi, j, Fj, n):
    if j<i:
        i, j = j, i
        Fi, Fj = Fj, Fi
  
    # Find the value of F(i + 1)
    # from F(i) and F(j)
    Fi1 = (Fj - fibonacci(j-i-1)*Fi)\
                      //fibonacci(j-i)
  
    # When n is equal to i
    if n == i:
        Fn = Fi
  
    # When n is greater than i 
    elif n>i:
        b = Fi
        Fn = Fi1
        n = n - 1
        while n != i:
            n = n - 1
            a = b
            b = Fn
            Fn = a + b
  
    # When n is smaller than i
    else:
        a = Fi
        b = Fi1
        while n != i:
            n = n + 1
            Fn = b - a
            b = a
            a = Fn
    return Fn
  
# Driver code
if __name__ == '__main__':
    i = 3
    Fi = 5
    j = -1
    Fj = 4
    n = 5
    print(determineFn(i, Fi, j, Fj, n))

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Output:

12


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