Recursive Sequences
Sequences where each term is a function of the previous terms.
This post is part of the book Introduction to Algorithms and Machine Learning: from Sorting to Strategic Agents. Suggested citation: Skycak, J. (2021). Recursive Sequences. In Introduction to Algorithms and Machine Learning: from Sorting to Strategic Agents. https://justinmath.com/recursivesequences/
A recursive sequence is a sequence where each term is a function of the previous terms.
For example, consider the following rule for generating a recursive sequence: starting with $3,$ generate each term by doubling the previous term and adding $1.$ The terms of this sequence are as follows:
Implementing Recursive Sequences
One way to implement recursive sequences in code is to store all terms in an array. For example, a function that returns the first $n$ terms of the recursive sequence “starting with $3,$ generate each term by doubling the previous term and adding $1$” could be implemented as follows:
def calc_first_n_terms(n):
terms = [3]
while len(terms) < n:
prev_term = terms[1]
next_term = 2 * prev_term + 1
terms.append(next_term)
return terms
If all we want is the $n$th term, then it can sometimes be more convenient to make the implementation itself recursive. This way, we don’t have to bother storing any intermediate values.
def calc_nth_term(n):
if n == 1:
return 3
else:
prev_term = calc_nth_term(n1)
return 2 * prev_term + 1
To understand how the function above works, first notice that calc_nth_term(1)
will return $3.$ This is called the base case. If a different input is passed, then the function will keep calling itself in a lower input until it reaches the base case.
calc_nth_term(4)

> prev_term
= calc_nth_term(3)

> prev_term
= calc_nth_term(2)

> prev_term
= calc_nth_term(1)

> return 3

<

< return 2 * 3 + 1 = 7

< return 2 * 7 + 1 = 15

< return 2 * 15 + 1 = 31
Output: 31
Exercises
Several different recursive sequences are described below. For each sequence, write a function to generate an array containing first $n$ terms, and then write a separate recursive function to generate the $n$th term. Be sure to work these sequences out by hand and write tests.
 Starting with $5,$ generate each term by multiplying the previous term by $3$ and subtracting $4.$
 Starting with $25,$ generate each term by taking half of the previous term if it's even, or multiplying by $3$ and adding $1$ if it's odd. (This is an instance of a Collatz sequence.)
 Starting with $0,1,$ generate each term by adding the previous two terms. (This is the famous Fibonacci sequence.)
 Starting with $2,3,$ generate each term by adding the product of the previous two terms.
This post is part of the book Introduction to Algorithms and Machine Learning: from Sorting to Strategic Agents. Suggested citation: Skycak, J. (2021). Recursive Sequences. In Introduction to Algorithms and Machine Learning: from Sorting to Strategic Agents. https://justinmath.com/recursivesequences/