Catalan numbers The Catalan numbers \(C_{n}\) are a sequence of integers \(1,1,2,5,14,42,132 \ldots\) that play an important role in quantum mechanics and the theory of disordered systems. (They were central to Eugene Wigner's proof of the so- called semicircle law.) They are given by $$ C_{0}=1, \quad C_{n+1}=\frac{4 n+2}{n+2} C_{n} . $$ Write a program that prints in increasing order all Catalan numbers less than or equal to one billion.

The Catalan numbers are an interesting sequence of integers that appear in various fields of mathematics. This sequence starts with 1 and continues as 1, 2, 5, 14, 42, 132, and so forth. These numbers are important because they often appear in combinatorial mathematics, and are used in problems related to tree structures, lattice paths, and even in quantum mechanics. The sequence is defined using a recurrence relation given by:
\] C_{0} = 1 \ \ \[ C_{n+1} = \dfrac{4n + 2}{n + 2} C_{n} \ \
Here, the next Catalan number, \( C_{n+1} \), is derived from the current Catalan number, \( C_n \). To compute these numbers iteratively, we start with \( C_0 \) and use the formula to find each successive number in the sequence.

To find multiple Catalan numbers, we can use iterative computation. This means computing each number in the sequence step-by-step using a loop.
Start by setting the initial value: \( C_0 = 1 \). Then, use a loop to compute the next values based on the recurrence formula. This prevents errors that might occur in manual calculations and ensures that we can quickly generate a large number of Catalan numbers.
For instance, using the given formula:
\] C_{n+1} = \dfrac{4n + 2}{n + 2} C_{n} \ \[
We compute each new number by substituting the current value of \( n \). Note that we adjust the index by incrementing \( n \) with each step of the loop, ensuring we generate the next number in the sequence. This process is repeated until the next number exceeds one billion, as specified in the problem.

To implement the solution programmatically, we need clear programming logic. Here's a structured approach:

• Initialize variables to store the first Catalan number, \( C_0 \), and a list to hold all computed Catalan numbers.
• Set up a loop to continuously compute new Catalan numbers.
• Use the given formula to calculate each subsequent number.
• Check if the next number exceeds one billion; if it does, stop the loop.
• Print out the computed values.

Here’s an example implementation in Python:
```python catalan_numbers = [] C_n = 1 n = 0 while C_n <= 1_000_000_000: catalan_numbers.append(C_n) C_n = (4*n + 2) * C_n // (n + 2) n += 1 print(catalan_numbers) ```
In this code, we initialize \( C_n \) to 1 and \( n \) to 0. We then enter a loop, adding each new Catalan number to our list and calculating the next using integer division to avoid floating-point inaccuracies. This basic structure ensures that we compute and display all Catalan numbers less than or equal to one billion.