Fibonacci Series
  1. What is Fibonacci Series
  2. Fibonacci Series Logic
  3. Fibonacci Series Formula
  4. Fibonacci Spiral
  5. Fibonacci series algorithm
  6. Fibonacci Series in Python
    a. Fibonacci Series Using loop
    b. Fibonacci Series using Recursion
    c. Fibonacci Series using Dynamic Programming

Leonardo Pisano Bogollo was an Italian mathematician from the Republic of Pisa and was considered the most talented Western mathematician of the Middle Ages. He lived between 1170 and 1250 in Italy. “Fibonacci” was his nickname, which roughly means “Son of Bonacci”. Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before!

What is Fibonacci Series?

Fibonacci Series is a pattern of numbers where each number is the result of addition of the previous two consecutive numbers. First 2 numbers start with 0 and 1. The third numbers in the sequence is 0+1=1. The 4th number is the addition of 2nd and 3rd number i.e. 1+1=2 and so on.
The Fibonacci Sequence is the series of numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …

Logic of Fibonacci Series

The next number is a sum of the two numbers before it.
The 3rd element is (1+0) = 1
The 4th element is (1+1) = 2
The 5th element is (2+1) = 3

Fibonacci Series Formula

Hence, the formula for calculating the series is as follows:
xn = xn-1 + xn-2 ; where
xn is term number “n”
xn-1 is the previous term (n-1)
xn-2 is the term before that

Fibonacci Spiral

An interesting property about these numbers is that when we make squares with these widths, we get a spiral. A Fibonacci spiral is a pattern of quarter-circles connected inside a block of squares with Fibonacci numbers written in each of the blocks. The number written in the bigger square is a sum of the next 2 smaller squares. This is a perfect arrangement where each block denoted a higher number than the previous two blocks. The main idea has been derived from the Logarithmic pattern which also looks similar. These numbers are also related to the golden ratio.

Fibonacci Series
Fibonacci Series

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Fibonacci Series Algorithm

Iterative Approach

  • Initialize variables a,b to 1
  • Initialize for loop in range[1,n) # n exclusive
  • Compute next number in series; total = a+b
  • Store previous value in b
  • Store total in a

Recursive Approach

  • If n equals 1 or 0; return 1
  • Else return fib(n-1) + fib(n-2)

Dynamic Programming Approach

  • Initialize an array arr of size n to zeros
  • If n equals 0 or 1; return 1 Else
  • Initalize arr[0] and arr[1] to 1
  • Run for loop in range[2,num]
  • Compute the value arr[I]=arr[I-1] +arr[I-2]
  • The array has the sequence computed till n

Hence, the solution would be to compute the value once and store it in an array from where it can be accessed the next time the value is required. Therefore, we use dynamic programming in such cases. The conditions for implementing dynamic programming are
1. overlapping sub-problems
2. optimal substructure

Fibonacci Series in Python

Iterative Approach


def fib_iter(n):
    a=1
    b=1
    if n==1:
        print('0')
    elif n==2:
        print('0','1')
    else:
        print("Iterative Approach: ", end=' ')
        print('0',a,b,end=' ')
        for i in range(n-3):
            total = a + b
            b=a
            a= total
            print(total,end=' ')
        print()
        return b
        
fib_iter(5)

TEST THE CODE

Output : Iterative Approach : 0 1 1 2 3

Recursive Approach


def fib_rec(n):
    if n == 1:
        return [0]
    elif n == 2:
        return [0,1]
    else:
        x = fib_rec(n-1)
        # the new element the sum of the last two elements
        x.append(sum(x[:-3:-1]))
        return x
x=fib_rec(5)
print(x)

TEST THE CODE

Output – 0, 1, 1, 2, 3

Dynamic Programming Approach

There is a slight modification to the iterative approach. We use an additional array.

def fib_dp(num):
    arr = [0,1]
    print("Dynamic Programming Approach: ",end= ' ')
    if num==1:
        print('0')
    elif num==2:
        print('[0,','1]')
    else:
        while(len(arr)<num):
            arr.append(0)
        if(num==0 or num==1):
            return 1
        else:
            arr[0]=0
            arr[1]=1
            for i in range(2,num):
                arr[i]=arr[i-1]+arr[i-2]
            print(arr)    
            return arr[num-2]
fib_dp(5)

TEST THE CODE

Output – 0, 1, 1, 2, 3

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