Hi! im looking to find relationships between sets of polygonal numbers.
I have found a few through the use of finite differences, (a relationship occurs between the formula to determine the polygonal number).
I have also found that the 2nd derivative of these formula corresponds to the polygonal number set. eg triangular the 2nd derivative is 1.
Any ideas would be appreciated! :):) thanks!
Formulas presented at this site may help you:
http://en.wikipedia.org/wiki/Polygonal_number
Hello! It's great that you're exploring relationships between sets of polygonal numbers. Finite differences and derivatives are indeed useful tools for investigating these relationships. Let me share a few more ideas that you can explore:
1. Generating Functions: One powerful technique is to use generating functions. You can create a generating function for a particular polygonal number set and then manipulate it to find relationships with other sets. The coefficients of terms in the generating function can reveal interesting connections.
2. Recurrence Relations: Many polygonal number sets can be defined using recursion. By studying the recurrence relations, you might uncover patterns and connections between different sets of polygonal numbers.
3. Catalan Numbers: The Catalan numbers are a sequence of natural numbers that have connections with various polygonal number sets. Exploring the relationships between these sets and the Catalan numbers can yield interesting findings.
4. Visual Patterns: Sometimes, visually examining polygonal numbers can lead to insights. Explore the geometric patterns created by different sets of polygonal numbers and look for commonalities or symmetries.
5. Number Theory: Polygonal numbers have connections to number theory. For example, certain sets of polygonal numbers correspond to triangular or square numbers. Investigate number theory concepts like divisibility, congruence, and factorization to find relationships.
Remember to apply numerical examples and test various polygonal number sets to observe any emerging patterns or connections. Experimentation and observation are key to uncovering relationships between sets of polygonal numbers. Good luck with your exploration!