Pascal's triangle patterns

Pascal's triangle is a triangular array of numbers that is named after the French mathematician Blaise Pascal. Each number in the triangle is the sum of the two numbers directly above it, as shown below:

Pascal's triangle contains many interesting number patterns. Here are some examples:

1. Fibonacci numbers: If you add up every other number in a row of Pascal's triangle, starting with the first number, you get the Fibonacci sequence. For example, in the fifth row, the numbers are 1 4 6 4 1, and if you add up every other number (1 + 6 + 1), you get the Fibonacci sequence 1 7 8.

2. Powers of 2: The sum of the numbers in each row of Pascal's triangle is equal to 2 raised to the power of the row number. For example, in the fourth row, the sum of the numbers is 1 + 3 + 3 + 1 = 8, which is equal to 2^4.

3. Triangular numbers: The nth number in the nth row of Pascal's triangle is equal to the nth triangular number. For example, the third number in the third row is 3, which is the third triangular number.

4. Binomial coefficients: The numbers in Pascal's triangle are also known as binomial coefficients because they can be used to expand the binomial (a + b)^n. For example, the coefficients in the fourth row of Pascal's triangle are 1 4 6 4 1, which correspond to the terms a^4 b^0, a^3 b^1, a^2 b^2, a^1 b^3, and a^0 b^4, respectively.

These are just a few of the many number patterns that can be found in Pascal's triangle. It is a fascinating mathematical object with many applications in combinatorics, probability theory, and number theory.