Heronian Triangle - Almost-equilateral Heronian Triangles

Almost-equilateral Heronian Triangles

Since the area of an equilateral triangle with rational sides is an irrational number, no equilateral triangle is Heronian. However, there is a unique sequence of Heronian triangles that are "almost equilateral" because the three sides are of the form n − 1, n, n + 1. The first few examples of these almost-equilateral triangles are listed in the following table (sequence A003500 in OEIS):

Subsequent values of n can be found by multiplying the previous value by 4, then subtracting the value prior to that one (52 = 4 × 14 − 4, 194 = 4 × 52 − 14, etc.), thus:

where t denotes any row in the table. This is a Lucas sequence.

The sequence of integer pairs (x, y) = (n/2, inradius) are solutions to the Pell's equation x2 − 3y2 = 1, which can in turn be derived from the regular continued fraction expansion for √3.

n is of the form, where k is 7, 97, 1351, 18817, …. The numbers in this sequence have the property that k consecutive integers have integral standard deviation.