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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 (list; graph; refs; listen; history; text; internal format)
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 19-20 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

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Last modified August 17 22:32 EDT 2018. Contains 313817 sequences. (Running on oeis4.)