NumPy – Fibonacci Series using Binet Formula

All of us are familiar with Fibonacci Series. Each number in the sequence is the sum of the two numbers that precede it. So, the sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34…… In this tutorial, we will implement the same using NumPy with the aid of Binet formula.

Binet Formula
  fn = \left [\left (\frac{1 + \sqrt{5}}{2}  \right )^{n} - \left (\frac{1 - \sqrt{5}}{2}  \right )^{n}  \right ]\ast \frac{1}{\sqrt{5}}  where, alpha = \left (\frac{1 + \sqrt{5}}{2}  \right ), beta = \left (\frac{1 - \sqrt{5}  }{2}\right )

n’ is the parameter which relates the first ‘n’ numbers of Fibonacci Series. In the first example we are going to findout first 10 numbers of Fibonacci Series (n = 10), after that we takes the parameter ‘n’ from user and produce the corresponding result.

NOTE : We are ignoring the first element(0) of Fibonacci Series

Example 1: To find first 10 Fibonacci numbers .



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import numpy as np
   
# We are creating an array contains n = 10 elements
# for getting first 10 Fibonacci numbers
a = np.arange(1, 11)
lengthA = len(a)
  
# splitting of terms for easiness
sqrtFive = np.sqrt(5)
alpha = (1 + sqrtFive) / 2
beta = (1 - sqrtFive) / 2
  
# Implementation of formula
# np.rint is used for rounding off to integer
Fn = np.rint(((alpha ** a) - (beta ** a)) / (sqrtFive))
print("The first {} numbers of Fibonacci series are {} . ".format(lengthA, Fn))

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Output :

The first 10 numbers of Fibonacci series are [ 1. 1. 2. 3. 5. 8. 13. 21. 34. 55.] .

Example 2 : To find first ‘n’ Fibonacci numbers ..

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import numpy as np
  
# We are creating an array contains n elements
# for getting first 'n' Fibonacci numbers
fNumber = int(input("Enter the value of n + 1'th number : "))
a = np.arange(1, fNumber)
length_a = len(a)
  
# splitting of terms for easiness
sqrt_five = np.sqrt(5)
alpha = (1 + sqrt_five) / 2
beta = (1 - sqrt_five) / 2
  
# Implementation of formula
# np.rint is used for rounding off to integer
Fn = np.rint(((alpha ** a) - (beta ** a)) / (sqrt_five))
print("The first {} numbers of Fibonacci series are {} . ".format(length_a, Fn))

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Output :

# Here user input was 10
Enter the value of n+1'th number :10
The first 9 numbers of Fibonacci series are [ 1.  1.  2.  3.  5.  8. 13. 21. 34.] . 

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