Solving the Recurrence Relation

Oct 13, 2025 by windailys.com 109 views

Introduction

In the realm of combinatorics, a Gessel walk is a type of walk on the square lattice that starts and ends at the origin. The possible steps in a Gessel walk are (1,0), (-1,0), (1,1), and (-1,-1). This type of walk is named after the mathematician Ira Gessel, who first introduced it in the 1990s. In this article, we will delve into the world of Gessel walks and explore the concept of counting them.

What are Gessel Walks?

A Gessel walk is a walk on the square lattice that starts and ends at the origin. The lattice is a grid of points, and the walk can move in four possible directions: right (1,0), left (-1,0), down (0,-1), and up (0,1). However, in the case of Gessel walks, the steps are restricted to (1,0), (-1,0), (1,1), and (-1,-1). This means that the walk can only move horizontally or diagonally.

Counting Gessel Walks

The problem of counting Gessel walks is a classic example of a combinatorial problem. The number of Gessel walks of length 2n is denoted by g(n). The sequence g(n) is a well-known sequence in the Online Encyclopedia of Integer Sequences (OEIS) with the identifier A135404.

The Recurrence Relation

The number of Gessel walks of length 2n can be calculated using a recurrence relation. Let g(n) be the number of Gessel walks of length 2n. Then, the recurrence relation is:

g(n) = 2g(n-1) + g(n-2)

This recurrence relation can be derived by considering the last step of the walk. If the last step is (1,0) or (-1,0), then the walk can be divided into two parts: the first part is a Gessel walk of length 2(n-1), and the second part is a single step. If the last step is (1,1) or (-1,-1), then the walk can be divided into two parts: the first part is a Gessel walk of length 2(n-2), and the second part is two steps.

The Initial Conditions

The initial conditions for the recurrence relation are g(0) = 1 and g(1) = 2. These initial conditions can be derived by considering the possible walks of length 0 and 1.

Solving the Recurrence Relation

The recurrence relation g(n) = 2g(n-1) + g(n-2) can be solved using various methods, including:

Iterative method: This method involves calculating the values of g(n) iteratively using the recurrence relation.
Recursive method: This method involves calculating the values of g(n) recursively using the recurrence relation.
Closed-form solution: This method involves finding a closed-form expression for g(n) using the recurrence relation.

Iterative Method

The iterative method involves calculating the values of g(n) iteratively using the recurrence relation. The values of g(n) can be calculated as follows:

g(0) = 1 g(1) = 2 g(2) = 2g(1) + g(0) = 2(2) + 1 = 5 g(3) = 2g(2) + g(1) = 2(5) + 2 = 12 g(4) = 2g(3) + g(2) = 2(12) + 5 = 29 g(5) = 2g(4) + g(3) = 2(29) + 12 = 70

Recursive Method

The recursive method involves calculating the values of g(n) recursively using the recurrence relation. The values of g(n) can be calculated as follows:

g(0) = 1 g(1) = 2 g(n) = 2g(n-1) + g(n-2)

Closed-Form Solution

The closed-form solution involves finding a closed-form expression for g(n) using the recurrence relation. The closed-form expression for g(n) is:

g(n) = (2^n + (-1)^n) * (n+1)

This closed-form expression can be derived using various methods, including:

Generating functions: This method involves using generating functions to find a closed-form expression for g(n).
Recurrence relations: This method involves using recurrence relations to find a closed-form expression for g(n).
Combinatorial methods: This method involves using combinatorial methods to find a closed-form expression for g(n).

Code Golf

Code golf is a popular programming challenge that involves writing the shortest possible code to solve a problem. In the context of Gessel walks, code golf involves writing the shortest possible code to calculate the number of Gessel walks of length 2n.

Example Code

Here is an example of code in Python that calculates the number of Gessel walks of length 2n:

def gessel_walks(n):
    if n == 0:
        return 1
    elif n == 1:
        return 2
    else:
        return 2*gessel_walks(n-1) + gessel_walks(n-2)

This code uses the recursive method to calculate the number of Gessel walks of length 2n.

Conclusion

In this article, we have explored the concept of Gessel walks and the problem of counting them. We have discussed the recurrence relation, initial conditions, and methods for solving the recurrence relation. We have also provided an example of code in Python that calculates the number of Gessel walks of length 2n. The problem of counting Gessel walks is a classic example of a combinatorial problem, and it has many applications in mathematics and computer science.

References

OEIS A135404: This is the Online Encyclopedia of Integer Sequences (OEIS) entry for the sequence g(n).
Gessel, I. M.: This is the paper by Ira Gessel that introduced the concept of Gessel walks.
Stanley, R. P.: This is the book by Richard P. Stanley that discusses combinatorial methods for solving recurrence relations.

Introduction

In our previous article, we explored the concept of Gessel walks and the problem of counting them. We discussed the recurrence relation, initial conditions, and methods for solving the recurrence relation. In this article, we will answer some frequently asked questions about Gessel walks and counting them.

Q: What is a Gessel walk?

A Gessel walk is a type of walk on the square lattice that starts and ends at the origin. The possible steps in a Gessel walk are (1,0), (-1,0), (1,1), and (-1,-1).

Q: How do I calculate the number of Gessel walks of length 2n?

You can calculate the number of Gessel walks of length 2n using the recurrence relation:

g(n) = 2g(n-1) + g(n-2)

where g(n) is the number of Gessel walks of length 2n.

Q: What are the initial conditions for the recurrence relation?

The initial conditions for the recurrence relation are g(0) = 1 and g(1) = 2.

Q: How do I solve the recurrence relation?

You can solve the recurrence relation using various methods, including:

Iterative method: This method involves calculating the values of g(n) iteratively using the recurrence relation.
Recursive method: This method involves calculating the values of g(n) recursively using the recurrence relation.
Closed-form solution: This method involves finding a closed-form expression for g(n) using the recurrence relation.

Q: What is the closed-form solution for g(n)?

The closed-form solution for g(n) is:

g(n) = (2^n + (-1)^n) * (n+1)

Q: How do I write code to calculate the number of Gessel walks of length 2n?

You can write code to calculate the number of Gessel walks of length 2n using various programming languages, including Python. Here is an example of code in Python that calculates the number of Gessel walks of length 2n:

def gessel_walks(n):
    if n == 0:
        return 1
    elif n == 1:
        return 2
    else:
        return 2*gessel_walks(n-1) + gessel_walks(n-2)

Q: What are some applications of Gessel walks?

Gessel walks have many applications in mathematics and computer science, including:

Combinatorics: Gessel walks are used to count and arrange objects in combinatorics.
Recurrence relations: Gessel walks are used to solve recurrence relations in mathematics.
Code golf: Gessel walks are used to write code to solve problems in code golf.

Q: Where can I find more information about Gessel walks?

You can find more information about Gessel walks in various resources, including:

Online Encyclopedia of Integer Sequences (OEIS): This is a website that provides information about integer sequences, including Gessel walks.
Mathematical papers: This is a website that provides information about mathematical papers, including papers about Gessel walks.
Books: This is a website that provides information about books, including books about combinatorics and recurrence relations.

Conclusion

In this article, we have answered some frequently asked questions about Gessel walks and counting them. We have discussed the recurrence relation, initial conditions, and methods for solving the recurrence relation. We have also provided an example of code in Python that calculates the number of Gessel walks of length 2n. We hope that this article has been helpful in answering your questions about Gessel walks.