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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.

%I A255011 #101 Aug 09 2025 19:38:58
%S A255011 0,4,56,340,1120,3264,6264,13968,22904,38748,58256,95656,120960,
%T A255011 192636,246824,323560,425408,587964,682296,932996,1061232,1327524,
%U A255011 1634488,2049704,2227672,2806036,3275800,3810088,4307520,5298768,5577096,6958848,7586496,8672520,9901352
%N 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.
%C A255011 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
%C A255011 a(n) is always divisible by 4, by symmetry. If n is odd, a(n) is divisible by 8.
%C A255011 From _Michael De Vlieger_, Feb 19-20 2015: (Start)
%C A255011 For n > 0, the vertices of the bounding square generate diametrical bisectors that cross at the center. Thus each diagram has fourfold symmetry.
%C A255011 For n > 0, an orthogonal n X n grid is produced by corresponding horizontal and vertical points on opposite sides.
%C A255011 Terms {1, 3, 9} are not congruent to 0 (mod 8).
%C A255011 Number of edges: {0, 8, 92, 596, 1936, 6020, 11088, 26260, 42144, 72296, 107832, ...}. See A331448. (End)
%H A255011 Zhao Hui Du, <a href="/A255011/b255011.txt">Table of n, a(n) for n = 0..136</a> (terms 0..52 from Lars Blomberg).
%H A255011 Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, <a href="http://neilsloane.com/doc/rose_5.pdf">Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids</a>, (2021); Also on <a href="https://arxiv.org/abs/2009.07918">arXiv</a>, arXiv:2009.07918 [math.CO], 2020.
%H A255011 Michael De Vlieger, <a href="/A255011/a255011.pdf">Diagrams of A255011(n) for n <= 10</a>
%H A255011 B. Poonen and M. Rubinstein (1998) <a href="http://math.mit.edu/~poonen/papers/ngon.pdf">The Number of Intersection Points Made by the Diagonals of a Regular Polygon</a>, SIAM J. Discrete Mathematics 11(1), pp. 135-156, doi:<a href="http://dx.doi.org/10.1137/S0895480195281246">10.1137/S0895480195281246</a>, arXiv:<a href="http://arXiv.org/abs/math.MG/9508209">math.MG/9508209</a> (has fewer typos than the SIAM version)
%H A255011 Scott R. Shannon, <a href="/A331452/a331452_6.png">Colored illustration for a(1)</a>
%H A255011 Scott R. Shannon, <a href="/A331452/a331452_12.png">Colored illustration for a(2)</a>
%H A255011 Scott R. Shannon, <a href="/A331452/a331452_1.png">Colored illustration for a(3)</a>
%H A255011 Scott R. Shannon, <a href="/A331452/a331452_21.png">Colored illustration for a(4)</a>
%H A255011 Scott R. Shannon, <a href="/A331452/a331452_24.png">Colored illustration for a(5)</a>
%H A255011 Scott R. Shannon, <a href="/A255011/a255011.png">Image for n = 2</a>.
%H A255011 Scott R. Shannon, <a href="/A255011/a255011_1.png">Image for n = 3</a>.
%H A255011 Scott R. Shannon, <a href="/A255011/a255011_2.png">Image for n = 4</a>.
%H A255011 Scott R. Shannon, <a href="/A255011/a255011_3.png">Image for n = 5</a>.
%H A255011 Scott R. Shannon, <a href="/A255011/a255011_4.png">Image for n = 10</a>.
%H A255011 N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 20.
%F A255011 No formula is presently known. - _N. J. A. Sloane_, Feb 04 2020
%e A255011 For n = 3, the perimeter of the square contains 12 points:
%e A255011   * * * *
%e A255011   *     *
%e A255011   *     *
%e A255011   * * * *
%e A255011 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.
%e A255011 For n = 1, the full picture is:
%e A255011   *-*
%e A255011   |X|
%e A255011   *-*
%e A255011 The lines form four triangular regions, so a(1) = 4.
%e A255011 For n = 0, the square can be regarded as consisting of a single point, producing no lines or polygons, and so a(0) = 0.
%Y A255011 Cf. A092098 (triangular analog), A331448 (edges), A331449 (points), A334699 (k-gons).
%Y A255011 For the circular analog see A006533, A007678.
%K A255011 nonn
%O A255011 0,2
%A A255011 _Johan Westin_, Feb 12 2015
%E A255011 a(11)-a(29) from _Hiroaki Yamanouchi_, Feb 23 2015
%E A255011 Offset changed by _N. J. A. Sloane_, Jan 23 2020