Areas of Triangles and other Polygons with Vertices from Various Sequences

TL;DR

This paper derives formulas for calculating areas of triangles and polygons with vertices from various integer sequences, including Fibonacci and Pell sequences, expanding geometric understanding of these sequences.

Contribution

It introduces new formulas for polygon areas with vertices from multiple integer sequences, generalizing previous specific cases.

Findings

Formulas for areas of polygons with Fibonacci and Pell sequence vertices

Extension of area formulas to various polygonal number sequences

Application to polygons with vertices from generalized Fibonacci sequences

Abstract

Motivated by Elementary Problem B-1172 in the Fibonacci Quarterly (vol. 53, no. 3, pg. 273), formulas for the areas of triangles and other polygons having vertices with coordinates taken from various sequences of integers are obtained. The sequences discussed are Polygonal number sequences as well as Fibonacci, Lucas, Jacobsthal, Jacobsthal-Lucas, Pell, Pell-Lucas, and Generalized Fibonacci sequences. The polygons have vertices with the form $(p_n,p_{n+k}),\,\, (p_{n+2k},p_{n+3k}),\,\,\dots ,(p_{n+(2m-2)k},p_{n+(2m-1)k)}$.