OFFSET
0,2
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.
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).
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).
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Noah Priluck (npriluck(AT)gmail.com), May 08 2007
EXTENSIONS
Edited by Michel Marcus, Jul 15 2013
STATUS
approved