Python Program for Zeckendorf\’s Theorem (Non-Neighbouring Fibonacci Representation)
Given a number, find a representation of number as sum of non-consecutive Fibonacci numbers.
Examples:
Input: n = 10 Output: 8 2 8 and 2 are two non-consecutive Fibonacci Numbers and sum of them is 10. Input: n = 30 Output: 21 8 1 21, 8 and 1 are non-consecutive Fibonacci Numbers and sum of them is 30.
The idea is to use Greedy Algorithm.
1) Let n be input number 2) While n >= 0 a) Find the greatest Fibonacci Number smaller than n. Let this number be 'f'. Print 'f' b) n = n - f
Python3
# Python program for Zeckendorf's theorem. It finds representation # of n as sum of non-neighbouring Fibonacci Numbers. # Returns the greatest Fibonacci Number smaller than # or equal to n. def nearestSmallerEqFib(n): # Corner cases if (n = = 0 or n = = 1 ): return n # Finds the greatest Fibonacci Number smaller # than n. f1, f2, f3 = 0 , 1 , 1 while (f3 < = n): f1 = f2; f2 = f3; f3 = f1 + f2; return f2; # Prints Fibonacci Representation of n using # greedy algorithm def printFibRepresntation(n): while (n> 0 ): # Find the greates Fibonacci Number smaller # than or equal to n f = nearestSmallerEqFib(n); # Print the found fibonacci number print (f,end = " " ) # Reduce n n = n - f # Driver code test above functions n = 30 print ( "Non-neighbouring Fibonacci Representation of" , n, "is" ) printFibRepresntation(n) |
Output:
Non-neighbouring Fibonacci Representation of 30 is 21 8 1
Please refer complete article on Zeckendorf’s Theorem (Non-Neighbouring Fibonacci Representation) for more details!