Let T be the equilaterall triangle with sides = 1 length.

Let $\displaystyle a_{n}$ be the # of circles that can be packed tightly in n rows inside the triangle.

$\displaystyle

a_{1} = 1

$

$\displaystyle

a_{2} = 3

$

$\displaystyle

a_{3} = 6

$

$\displaystyle

a_{4} = 10

$

Let $\displaystyle A_{n}$ be the combined area of the $\displaystyle a_{n}$ circles. Find the limit of $\displaystyle A_{n}$ as n -> infinity.

I believe I have to find a series for the number of circles in the triangles, right? So I need to find something to represent 1, 3, 6, 10, 15 ... ?