# Generalized Fibonacci Numbers

We all know that Fibonacci numbers (Fn) is defined by the recurrence relation

Fibonacci Numbers (Fn) = F(n-1) + F(n-2)
with seed values
F0 = 0 and F1 = 1

Similarly, we can generalise these numbers. Such number sequence is known as Generalized Fibonacci number (G)
Generalized Fibonacci number (G) is defined by the recurrence relation

Generalised Fibonacci Numbers (Gn) = (c * G(n-1)) + (d * G(n-2))
with seed values
G0 = a and G1 = b

### Finding Nth term

Given the four constant values of Generalised Fibonacci Numbers as a, b, c and d and an integer N, the task is to find the Nth term of the Generalised Fibonacci Numbers, i.e. Gn.
Examples:

Input: N = 2, a = 0, b = 1, c = 2, d = 3
Output:
Explanation:
As a = 0 -> G(0) = 0
b = 1 -> G(1) = 1
So, G(2) = 2 * G(1) + 3 * G(0) = 2
Input: N = 3, a = 0, b = 1, c = 2, d = 3
Output:

Naive Approach: Using the given values, find each term of the series till Nth term and then print the Nth term.
Time Complexity: O(2N)
Another Approach: The idea is to use DP tabulation to find all the terms till Nth terms and then print the Nth term.
Time Complexity: O(N)

Efficient Approach: Using matrix multiplication we can solve the given problem in log(N) time. Below is the implementation of the above approach:

## C++

 // C++ program to implement the   // Generalised Fibonacci numbers      #include    using namespace std;      // Helper function that multiplies   // 2 matrices F and M of size 2*2,   // and puts the multiplication   // result back to F[][]   void multiply(int F, int M);      // Helper function that calculates F[][]   // raised to the power N   // and puts the result in F[][]   void power(int F, int N, int m, int n);      // Function to find the Nth term   int F(int N, int a, int b, int m, int n)   {       // m 1       // n 0       int F = { { m, 1 }, { n, 0 } };          if (N == 0)           return a;          if (N == 1)           return b;          if (N == 2)           return m * b + n * a;          int initial           = { { m * b + n * a, b },               { b, a } };          power(F, N - 2, m, n);          // Discussed above      multiply(initial, F);          return F;   }      // Function that multiplies   // 2 matrices F and M of size 2*2,   // and puts the multiplication   // result back to F[][]   void multiply(int F, int M)   {       int x = F * M + F * M;       int y = F * M + F * M;       int z = F * M + F * M;       int w = F * M + F * M;          F = x;       F = y;       F = z;       F = w;   }      // Function that calculates F[][]   // raised to the power N   // and puts the result in F[][]   void power(int F, int N, int m, int n)   {       int i;       int M = { { m, 1 }, { n, 0 } };          for (i = 1; i <= N; i++)           multiply(F, M);   }      // Driver code   int main()   {       int N = 2, a = 0, b = 1, m = 2, n = 3;       printf("%d\n", F(N, a, b, m, n));          N = 3;       printf("%d\n", F(N, a, b, m, n));          N = 4;       printf("%d\n", F(N, a, b, m, n));          N = 5;       printf("%d\n", F(N, a, b, m, n));          return 0;   }

## Java

 // Java program to implement the   // Generalised Fibonacci numbers   import java.util.*;     class GFG{     // Function to find the Nth term   static int F(int N, int a, int b,                int m, int n)   {       // m 1       // n 0       int[][] F = { { m, 1 }, { n, 0 } };          if (N == 0)           return a;       if (N == 1)           return b;       if (N == 2)           return m * b + n * a;          int[][] initial = { { m * b + n * a, b },                           { b, a } };                                  power(F, N - 2, m, n);          // Discussed below       multiply(initial, F);          return F;   }      // Function that multiplies   // 2 matrices F and M of size 2*2,   // and puts the multiplication   // result back to F[][]   static void multiply(int[][] F, int[][] M)   {       int x = F * M +               F * M;       int y = F * M +               F * M;       int z = F * M +               F * M;       int w = F * M +               F * M;          F = x;       F = y;       F = z;       F = w;   }      // Function that calculates F[][]   // raised to the power N   // and puts the result in F[][]   static void power(int[][] F, int N,                     int m, int n)   {       int i;       int[][] M = { { m, 1 }, { n, 0 } };          for(i = 1; i <= N; i++)           multiply(F, M);   }      // Driver code  public static void main(String[] args)  {      int N = 2, a = 0, b = 1, m = 2, n = 3;       System.out.println(F(N, a, b, m, n));             N = 3;       System.out.println(F(N, a, b, m, n));              N = 4;       System.out.println(F(N, a, b, m, n));              N = 5;       System.out.println(F(N, a, b, m, n));   }  }     // This code is contributed by offbeat

## Python3

 # Python3 program to implement the  # Generalised Fibonacci numbers     # Function to find the Nth term  def F(N, a, b, m, n):             # m 1       # n 0       F = [[ m, 1 ], [ n, 0 ]]         if(N == 0):          return a      if(N == 1):          return b      if(N == 2):          return m * b + n * a         initial = [[ m * b + n * b, b ],                             [ b, a ]]         power(F, N - 2, m, n)         multiply(initial, F)         return F     # Function that multiplies  # 2 matrices F and M of size 2*2,  # and puts the multiplication  # result back to F[][]  def multiply(F, M):             x = (F * M +           F * M)         y = (F * M +          F * M)         z = (F * M +           F * M)         w = (F * M +           F * M)         F = x      F = y      F = z      F = w     # Function that calculates F[][]  # raised to the power N  # and puts the result in F[][]  def power(F, N, m, n):         M = [[ m, 1 ], [ n, 0 ]]      for i in range(1, N + 1):          multiply(F, M)     # Driver code  if __name__ == '__main__':         N, a, b, m, n = 2, 0, 1, 2, 3     print(F(N, a, b, m, n))         N = 3     print(F(N, a, b, m, n))         N = 4     print(F(N, a, b, m, n))         N = 5     print(F(N, a, b, m, n))     # This code is contributed by Shivam Singh

Output:

2
7
20
61


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Improved By : SHIVAMSINGH67, offbeat