A255011 Number of polygons formed by connecting all the 4n points on the perimeter of an n X n square by straight lines; a(0) = 0 by convention.

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, 5577096, 6958848, 7586496, 8672520, 9901352

Offset: 0

Author

Johan Westin, Feb 12 2015

Keywords

• nonn

Comments

There are n+1 points on each side of the square, but that counts the four corners twice, so there are a total of 4n points on the perimeter. - N. J. A. Sloane, Jan 23 2020

a(n) is always divisible by 4, by symmetry. If n is odd, 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 the 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, ...}. See A331448. (End)

Examples

			For n = 3, the perimeter of the square contains 12 points:
  * * * *
  *     *
  *     *
  * * * *
Connect each point to every other point with a straight line inside the square. Then count the polygons (or regions) that have formed. There are 340 polygons, so a(3) = 340.
For n = 1, the full picture is:
  *-*
  |X|
  *-*
The lines form four triangular regions, so a(1) = 4.
For n = 0, the square can be regarded as consisting of a single point, producing no lines or polygons, and so a(0) = 0.
		

Links

• Zhao Hui Du, Table of n, a(n) for n = 0..136 (terms 0..52 from Lars Blomberg).
• Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021); Also on arXiv, arXiv:2009.07918 [math.CO], 2020.
• Michael De Vlieger, Diagrams of A255011(n) for n <= 10
• B. Poonen and M. Rubinstein (1998) The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11(1), pp. 135-156, doi:10.1137/S0895480195281246, arXiv:math.MG/9508209 (has fewer typos than the SIAM version)
• Scott R. Shannon, Colored illustration for a(1)
• Scott R. Shannon, Colored illustration for a(2)
• Scott R. Shannon, Colored illustration for a(3)
• Scott R. Shannon, Colored illustration for a(4)
• Scott R. Shannon, Colored illustration for a(5)
• Scott R. Shannon, Image for n = 2.
• Scott R. Shannon, Image for n = 3.
• Scott R. Shannon, Image for n = 4.
• Scott R. Shannon, Image for n = 5.
• Scott R. Shannon, Image for n = 10.
• N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 20.

Crossrefs

Cf. A092098 (triangular analog), A331448 (edges), A331449 (points), A334699 (k-gons).

For the circular analog see A006533, A007678.

Formula

No formula is presently known. - N. J. A. Sloane, Feb 04 2020

Extensions

a(11)-a(29) from Hiroaki Yamanouchi, Feb 23 2015

Offset changed by N. J. A. Sloane, Jan 23 2020