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

Offset

0,2

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)

Links

Lars Blomberg, Table of n, a(n) for n = 0..52

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.

Formula

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

Example

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.

Crossrefs

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

For the circular analog see A006533, A007678.

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

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

Status

approved