It is easily proved that no equilateral triangle exists with integral length sides and integral area. However, the <dfn>almost equilateral triangle</dfn> 5-5-6 has an area of 12 square units.
We shall define an <dfn>almost equilateral triangle</dfn> to be a triangle for which two sides are equal and the third differs by no more than one unit.
Find the sum of the perimeters of all <dfn>almost equilateral triangle</dfn> with integral side lengths and area and whose perimeters do not exceed one billion (1,000,000,000).