Feat : Pascal's_Traingle

[MeKaustubh07]

🟨 Pascal’s Triangle Generator (R)

This script implements Pascal’s Triangle — a fundamental structure in combinatorics and number theory — using R.
Each entry in the triangle represents a binomial coefficient C(n, k) and satisfies the recursive property:

[
C(n, k) = C(n-1, k-1) + C(n-1, k)
]

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📘 Features

• Generate Pascal’s Triangle up to n rows
• Get a specific row or element (C(n, k))
• Print the triangle neatly (centered or left-aligned)
• Compute row sums (powers of 2) and diagonal sums (Fibonacci sequence)
• Visualize the Sierpiński Triangle pattern of odd numbers
• Return the triangle as a matrix representation
• Demonstrate properties and applications with examples

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⚙️ Functions Overview

Function	Description
pascal_triangle(n)	Generate full triangle up to n rows
pascal_row(n)	Get the nth row (0-indexed)
pascal_element(n, k)	Get a specific element C(n, k)
print_pascal_triangle(n, centered = TRUE)	Print formatted triangle
pascal_row_sum(n)	Get sum of row n = 2^n
pascal_all_row_sums(n)	Vector of row sums up to n
pascal_diagonal_sums(n)	Compute diagonal sums (Fibonacci pattern)
pascal_triangle_matrix(n)	Return matrix form of triangle
pascal_odd_pattern(n)	Return odd/even pattern (Sierpiński triangle)

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🧩 Example Usage

source("pascal_triangle.r")

# Generate Pascal's Triangle (10 rows)
print_pascal_triangle(10)

# Get row 7
pascal_row(7)

# Compute C(10, 3)
pascal_element(10, 3)

# Show Fibonacci diagonal sums
pascal_diagonal_sums(10)

# Display odd-number pattern (Sierpiński triangle)
odd_pattern <- pascal_odd_pattern(8)
print(odd_pattern)

[Copilot]

Pull Request Overview

Adds a new mathematics algorithm implementation for Pascal’s Triangle in R, including generation, element access, printing, derived properties, and visualizations.

• Introduces core functions: pascal_triangle, pascal_row, pascal_element, and helpers for sums, matrix, and odd-number pattern.
• Adds a formatted printer and a comprehensive (disabled) examples block demonstrating properties and usage.

[Copilot]

Pull Request Overview

Copilot reviewed 1 out of 1 changed files in this pull request and generated 6 comments.

Comments suppressed due to low confidence (1)

mathematics/pascal_triangle.r:1

• These are central binomial coefficients, not Catalan numbers. Either relabel the example as 'Central binomial coefficients' or compute Catalan numbers using choose(2n, n)/(n + 1).

# pascal_triangle.r

[Copilot]

Pull Request Overview

Copilot reviewed 1 out of 1 changed files in this pull request and generated 8 comments.

[Copilot]

Pull Request Overview

Copilot reviewed 1 out of 1 changed files in this pull request and generated 3 comments.