A128828 The number of regular hexagons found by constructing n equally-spaced points on each side of the hexagon and drawing lines parallel to the hexagon side.

1, 2, 15, 28, 65, 120

Offset: 0

Author

Noah Priluck (npriluck(AT)gmail.com), May 08 2007

Keywords

• nonn
• more

Comments

Use 1 midpoint (resp. 2 points) on each side placed to divide each side into 2 (resp. 3) equally-sized segments (or so on), do the same construction for every side of the hexagon so that each side is equally divided in the same way. Connect all such points to each other with lines that are parallel to at least 1 side of the hexagon.

Similar constructions to sequences A002717 (dividing a triangle), A000330 (dividing a square) and other sequences pending for similar constructions in other polygons.

Examples

			With 1 point (a midpoint on each side), 2 regular hexagons are found. With 3 points on each side, 15 regular hexagons are found in total and so on.
		

Links

• Noah Priluck, On Counting Regular Polygons Formed by Special Families of Parallel Lines, Geombinatorics Quarterly, Vol XVII (4), 2008, pp. 166-171. (note there is no document to download, see A128127 for pdf file).

Crossrefs

Cf. A128127, A128153.

Formula

a(n) = (11*n^4 + 78*n^3 + 1413*n^2 - 2322*n + 324)/324 when n = 3k,

(-13*n^4 + 670*n^3 - 3219*n^2 + 9934*n - 6724)/324 when n = 1 + 3k,

(5*n^4-46*n^3 + 1515*n^2 - 6046*n + 7940)/108 when n = 2 + 3k (conjecture).

Extensions

Edited by Michel Marcus, Jul 15 2013