

A255011


The number of polygons formed by connecting the points on the outline of an n X n square by straight lines.


1



0, 4, 56, 340, 1120, 3264, 6264, 13968, 22904, 38748, 58256, 95656, 120960, 192636, 246824, 323560, 425408, 587964, 682296, 932996, 1061232, 1327524, 1634488, 2049704, 2227672, 2806036, 3275800, 3810088, 4307520, 5298768
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OFFSET

1,2


COMMENTS

a(n) is always divisible by 4, due to symmetry. For every odd n, a(n) is divisible by 8.
From Michael De Vlieger, Feb 1920 2015: (Start)
For n > 0, the vertices of the bounding square generate diametrical bisectors that cross at center. Thus each diagram has fourfold symmetry.
For n > 0, an orthogonal n X n grid is produced by corresponding horizontal and vertical points on opposite sides.
Terms {1, 3, 9} are not congruent to 0 (mod 8).
Number of edges: {0, 8, 92, 596, 1936, 6020, 11088, 26260, 42144, 72296, 107832, ...}. (End)


LINKS

Table of n, a(n) for n=1..30.
Michael De Vlieger, Diagrams of A255011(n) for n <= 10


EXAMPLE

For n = 4, make an outline of points arranged in a 4 X 4 square:
* * * *
* *
* *
* * * *
Connect each point to every other point with a straight line. Once all the points are interconnected, count the polygons that have formed. For a 4 X 4 square, 340 polygons are formed, therefore a(4) = 340.
For n = 2, the full picture is:
**
X
**
The lines form four triangular regions, therefore a(2) = 4.
For n = 1, the square consists of a single point, producing no lines or polygons, and as such, a(1) = 0.


CROSSREFS

Cf. A092098 (Triangle).
Sequence in context: A006592 A201448 A195577 * A201620 A204108 A077122
Adjacent sequences: A255008 A255009 A255010 * A255012 A255013 A255014


KEYWORD

nonn,more


AUTHOR

Johan Westin, Feb 12 2015


EXTENSIONS

a(12)a(30) from Hiroaki Yamanouchi, Feb 23 2015


STATUS

approved



