  The article presents the mathematical sequences describing circle packing
densities in four different geometric configurations involving a hexagonal
lattice based equal circle packing in the Euclidian plane. The calculated
sequences take form of either polynomials or rational functions. If the circle
packing area is limited with a circle, the packing densities tend to decrease
with increasing number of the packed circles and converge to values lower than
{\pi}/(2\sqrt{3}). In cases with packing areas limited by equilateral triangles
or equilateral hexagons the packing densities tend to increase with increasing
number of the packed circles and converge to {\pi}/(2\sqrt{3}). The equilateral
hexagons are shown to be the preferred equal circle packing surface areas with
practical applications searching for high equal circle packing densities, since
the packing densities with circle packing inside equilateral hexagons converge
faster to {\pi}/(2\sqrt{3}) than in the case of equilateral triangle packing
surface areas.
