Lucas numbers are similar to Fibonacci numbers. Lucas numbers are also defined as the sum of its two immediately previous terms. But here the first two terms are 2 and 1 whereas in Fibonacci numbers the first two terms are 0 and 1 respectively.
Mathematically, Lucas Numbers may be defined as:
The Lucas numbers are in the following integer sequence:
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123 …………..
Write a function int lucas(int n) n as argument and returns the n’th Lucas number.
Examples :
Input : 3 Output : 4 Input : 7 Output : 29
Method 1 ( Recursive Solution )
Below is recursive implementation based on simple recursive formula.
C/C++
// Recursive C/C++ program // to find n'th Lucas number #include <stdio.h> // recursive function int lucas(int n) { // Base cases if (n == 0) return 2; if (n == 1) return 1; // recurrence relation return lucas(n - 1) + lucas(n - 2); } // Driver Code int main() { int n = 9; printf("%d", lucas(n)); return 0; } |
Java
// Recursive Java program to // find n'th Lucas number class GFG { // recursive function public static int lucas(int n) { // Base cases if (n == 0) return 2; if (n == 1) return 1; // recurrence relation return lucas(n - 1) + lucas(n - 2); } // Driver Code public static void main(String args[]) { int n = 9; System.out.println(lucas(n)); } } // This code is contributed // by Nikita Tiwari. |
Python3
# Recursive Python 3 program # to find n'th Lucas number # recursive function def lucas(n) : # Base cases if (n == 0) : return 2 if (n == 1) : return 1 # recurrence relation return lucas(n - 1) + lucas(n - 2) # Driver code n = 9print(lucas(n)) # This code is contributed by Nikita Tiwari. |
C#
// Recursive C# program to // find n'th Lucas number using System; class GFG { // recursive function public static int lucas(int n) { // Base cases if (n == 0) return 2; if (n == 1) return 1; // recurrence relation return lucas(n - 1) + lucas(n - 2); } // Driver program public static void Main() { int n = 9; Console.WriteLine(lucas(n)); } } // This code is contributed by vt_m. |
PHP
<?php // Recursive php program to // find n'th Lucas number // recursive function function lucas($n) { // Base cases if ($n == 0) return 2; if ($n == 1) return 1; // recurrence relation return lucas($n - 1) + lucas($n - 2); } // Driver Code $n = 9; echo lucas($n); // This code is contributed by ajit. ?> |
Output :
76
Method 2 ( Iterative Solution )
The time complexity of above implementation is exponential. We can optimize it to work in O(n) time using iteration.
C/C++
// Iterative C/C++ program // to find n'th Lucas Number #include <stdio.h> // Iterative function int lucas(int n) { // declaring base values // for positions 0 and 1 int a = 2, b = 1, c, i; if (n == 0) return a; // generating number for (i = 2; i <= n; i++) { c = a + b; a = b; b = c; } return b; } // Driver Code int main() { int n = 9; printf("%d", lucas(n)); return 0; } |
Java
// Iterative Java program to // find n'th Lucas Number class GFG { // Iterative function static int lucas(int n) { // declaring base values // for positions 0 and 1 int a = 2, b = 1, c, i; if (n == 0) return a; // generating number for (i = 2; i <= n; i++) { c = a + b; a = b; b = c; } return b; } // Driver Code public static void main(String args[]) { int n = 9; System.out.println(lucas(n)); } } // This code is contributed // by Nikita tiwari. |
Python3
# Iterative Python 3 program # to find n'th Lucas Number # Iterative function def lucas(n) : # declaring base values # for positions 0 and 1 a = 2 b = 1 if (n == 0) : return a # generating number for i in range(2, n + 1) : c = a + b a = b b = c return b # Driver Code n = 9print(lucas(n)) # This code is contributed # by Nikita tiwari. |
C#
// Iterative C# program to // find n'th Lucas Number using System; class GFG { // Iterative function static int lucas(int n) { // declaring base values // for positions 0 and 1 int a = 2, b = 1, c, i; if (n == 0) return a; // generating number for (i = 2; i <= n; i++) { c = a + b; a = b; b = c; } return b; } // Driver Code public static void Main() { int n = 9; Console.WriteLine(lucas(n)); } } // This code is contributed by vt_m. |
PHP
<?php // Iterative php program // to find n'th Lucas Number function lucas($n) { // declaring base values // for positions 0 and 1 $a = 2; $b = 1; $c; $i; if ($n == 0) return $a; // generating number for ($i = 2; $i <= $n; $i++) { $c = $a + $b; $a = $b; $b = $c; } return $b; } // Driver Code $n = 9; echo lucas($n); // This code is contributed by ajit ?> |
Output :
76
References:
https://en.wikipedia.org/wiki/Lucas_number
This article is contributed by Harsh Agarwal. 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.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.
