blaise pascal is a mathematician who loves pattern. he discovered that in this special pattern of numbers, each row starts and ends with no. 1 and every number x is the sum of two numbers in the previous row closes to that number x. another pattern that can be observed is that the sum of terms in every row is

Certainly! You are describing Pascal's Triangle , a famous mathematical pattern discovered by Blaise Pascal. Here’s a detailed explanation:

Pascal's Triangle

Structure:

• Each row in Pascal’s Triangle starts and ends with the number 1.
• Every other number in the row is obtained by adding the two numbers directly above it from the previous row.

For example, the first few rows look like this:

      1
     1 1
    1 2 1
   1 3 3 1
  1 4 6 4 1

How the numbers are formed:

• The number at position k in row n (both starting from 0) is the sum of the numbers at positions k-1 and k in row n-1.
• If there is no number at those positions (like at the edges), it is considered 0.

Sum of terms in each row:

• Another interesting pattern is that the sum of the numbers in each row is a power of 2.
• Specifically, the sum of the numbers in the nth row is 2n2^n2n.

For example:

• Row 0: 1 → sum = 1 = 202^020
• Row 1: 1 + 1 = 2 = 212^121
• Row 2: 1 + 2 + 1 = 4 = 222^222
• Row 3: 1 + 3 + 3 + 1 = 8 = 232^323

Summary:

• Pascal’s Triangle is a triangular array of numbers.
• Each number is the sum of the two numbers above it.
• Rows start and end with 1.
• The sum of the numbers in the nth row is 2n2^n2n.

If you want, I can also help you generate Pascal’s Triangle programmatically or explore more patterns related to it!