What question did this study set out to answer?

This paper aims to demonstrate how recursive sequences, including Jacobsthal numbers, can be represented as weighted diagonal sums in Pascal's triangle using visual arguments.

June 27, 2026

A Visual Approach to Weighted Diagonal Sums in Pascal’s Triangle

Key Points

This paper aims to demonstrate how recursive sequences, including Jacobsthal numbers, can be represented as weighted diagonal sums in Pascal's triangle using visual arguments.
Utilized visual arguments based on Pascal's rule for binomial coefficients.
Analyzed the relationships between Fibonacci and Jacobsthal numbers through diagonal sums in Pascal’s triangle.
Proved properties of recursive sequences leveraging the visual representation.
Showed that Jacobsthal numbers can be expressed as weighted diagonal sums in Pascal's triangle.
Established new visual proofs for properties of Fibonacci and Jacobsthal sequences.
Provided further insights into the relationships between visual representations and recursive sequences.

Abstract

SummaryIt is well known that Fibonacci numbers appear as diagonal sums in Pascal’s triangle. In this paper, we explore how visual arguments based on Pascal’s rule for binomial coefficients can be used to show that other recursive sequences, such as the Jacobsthal numbers, can also be expressed as weighted diagonal sums in Pascal’s triangle. Furthermore, we apply this visual representation to prove properties of such sequences.

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A Visual Approach to Weighted Diagonal Sums in Pascal’s Triangle

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