A331448 -id:A331448 - cp's OEIS frontend

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

Johan Westin, Feb 12 2015

nonn

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)

			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.

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.

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

For the circular analog see A006533, A007678.

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

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

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

A331449 a(0) = 1 by convention; for n>0, a(n) is the number of points in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

Original entry on oeis.org

1, 5, 37, 257, 817, 2757, 4825, 12293, 19241, 33549, 49577, 87685, 101981, 178465, 220113, 286357, 379097, 551669, 606241, 880293, 951445, 1209049, 1507521, 1947877, 2002669, 2624409, 3056093, 3550425, 3955049, 5069037, 5062153, 6669665, 7081969, 8143193, 9365089, 10296469

Offset: 0

N. J. A. Sloane, Jan 23 2020

nonn

Equivalently, this is A334690(n) + 4*n.

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 arXiv:2009.07918.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 20.
Scott R. Shannon, Image for n = 1.
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.

Cf. A255011, A331448.

For the circular analog see A006533, A007678, A007569, A135565.

By Euler's formula, a(n) = A331448(n) - A255011(n) + 1.

a(11)-a(35) from Giovanni Resta, Jan 28 2020

A331454 Triangle read by rows: T(n,m) (n >= m >= 1) = number of line segments formed by drawing the lines connecting any two of the 2*(m+n) perimeter points of an m X n grid of squares.

Original entry on oeis.org

8, 28, 92, 80, 240, 596, 178, 508, 1028, 1936, 372, 944, 2004, 3404, 6020, 654, 1548, 3018, 4962, 8064, 11088, 1124, 2520, 4808, 7734, 12708, 17022, 26260, 1782, 3754, 6704, 10840, 16608, 22220, 32794, 42144, 2724, 5392, 9780, 14620, 22788, 30238, 44028, 54024, 72296, 3914, 7528, 12720, 19428, 29914, 37848, 54612, 67590, 86906, 107832

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 27 2020

Take a grid of m+1 X n+1 points. There are 2*(m+n) points on the perimeter. Join every pair of the perimeter points by a line (of finite length). The lines do not extend outside the grid. T(m,n) is the number of line segments formed when these lines intersect each other, and A331452(m,n) and A331453(m,n) give the number of regions and the number of vertices respectively.

For illustrations see the links in A331452.

			Triangle begins:
8,
28, 92,
80, 240, 596,
178, 508, 1028, 1936,
372, 944, 2004, 3404, 6020,
654, 1548, 3018, 4962, 8064, 11088,
1124, 2520, 4808, 7734, 12708, 17022, 26260,
1782, 3754, 6704, 10840, 16608, 22220, 32794, 42144,
2724, 5392, 9780, 14620, 22788, 30238, 44028, 54024, 72296,
...

Lars Blomberg, Table of n, a(n) for n = 1..703 (the first 37 rows)
N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.

The main diagonal is A331448.
The first two columns are A331757 and A331765.
Cf. A288187, A331452, A331453.

A334690 a(n) is the number of points in the interior of the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

Original entry on oeis.org

1, 29, 245, 801, 2737, 4801, 12265, 19209, 33513, 49537, 87641, 101933, 178413, 220057, 286297, 379033, 551601, 606169, 880217, 951365, 1208965, 1507433, 1947785, 2002573, 2624309, 3055989, 3550317, 3954937, 5068921, 5062033, 6669541, 7081841, 8143061, 9364953, 10296329, 10918073, 13772225, 14842297, 16312317

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 17 2020

nonn

a(n) = A331449(n) - 4*n. Computed from Lars Blomberg's b-file for A331449.

N. J. A. Sloane, Table of n, a(n) for n = 1..52

Cf. A255011, A331448, A331449.

These are the sums of the rows in A334691.

A367276 Place n equally spaced points on each side of a square, and join each of these points by a chord to the 3*n new points on the other three sides: sequence gives number of vertices in the resulting planar graph.

Original entry on oeis.org

4, 9, 69, 345, 1337, 2885, 7445, 12833, 23365, 36589, 64669, 80133, 138313, 176885, 233765, 312013, 455273, 513277, 741965, 819589, 1046245, 1310761, 1692961, 1772097, 2315289, 2713997, 3165125, 3552753, 4538845, 4602985, 6015561, 6432681, 7421345, 8550485, 9439621, 10063993, 12635769

Offset: 0

Scott R. Shannon, Nov 11 2023

nonn

We start with the four corner points of the square, and add n further points along each edge. Including the corner points, we end up with n+2 points along each edge, and the edge is divided into n+1 line segments.

Each of the n points added to an edge is joined by 3*n chords to the points that were added to the other three edges. There are 6*n^2 chords.

Scott R. Shannon, Image for n = 1.
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.

Cf. A367277 (interior vertices), A367278 (regions), A367279 (edges).
If the 4*n points are placed "in general position" instead of uniformly, we get sequences A334698, A367121, A367122.
If the 4*n points are placed uniformly and we also draw chords from the four corner points of the square to these 4*n points, we get A255011, A331448, A331449, A334690.

a(n) = A367279(n) - A367278(n) + 1 (Euler).

A345650 Number of polygon edges formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

Original entry on oeis.org

0, 8, 140, 1032, 3608, 11308, 21892, 51000, 83404, 143900, 217728, 367132, 456008, 749328, 952236, 1251056, 1649708, 2327232, 2653900, 3717444, 4158448, 5243680, 6488208, 8241988, 8780976, 11235028, 13116156

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Jun 21 2021

See A345459 for images of the polygons.

Cf. A331448 (number inside the square), A345459 (number of polygons), A345649 (number of vertices), A255011, A345648, A344993, A344857, A092098, A007678.

a(n) = A345459(n) + A345649(n) - 1.

A367122 Place n points in general position on each side of a square, and join every pair of the 4*n+4 boundary points by a chord; sequence gives number of edges in the resulting planar graph.

Original entry on oeis.org

8, 124, 780, 2816, 7480, 16428, 31724, 55840, 91656, 142460, 211948, 304224, 423800, 575596, 764940, 997568, 1279624, 1617660, 2018636, 2489920, 3039288, 3674924, 4405420, 5239776, 6187400, 7258108, 8462124, 9810080, 11313016, 12982380, 14830028, 16868224, 19109640, 21567356

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Nov 05 2023

nonn

"In general position" implies that the internal lines (or chords) only have simple intersections. There is no interior point where three or more chords meet.

See A334698 and A367121 for images of the square.

Cf. A334698 (vertices), A367121 (regions), A331448, A367119.

Conjecture: a(n) = 17*n^4 + 38*n^3 + 37*n^2 + 24*n + 8.

a(n) = A334698(n+1) + A367121(n) - 1 by Euler's formula.

A358409 Number of edges formed in a square by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on each of the two adjacent edges of the square.

Original entry on oeis.org

4, 12, 68, 316, 1020, 2524, 5420, 10348, 18044, 29244, 45940, 66188, 97796, 135772, 182532, 240932, 321612, 405852, 525184, 646088, 796388, 974740, 1199244, 1407140, 1700944, 2004576, 2356296, 2729256, 3221296, 3630296, 4272656, 4835984, 5522768, 6269016, 7084056, 7835068, 8987192, 10005400

Offset: 1

Scott R. Shannon, Nov 14 2022

nonn

See A358407 and A358408 for images of the square.

Cf. A358407 (regions), A358408 (vertices), A355800, A331448.

a(n) = A358407(n) + A358408(n) - 1 by Euler's formula.

A367324 Table read by antidiagonals: Place k equally spaced points on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives the number of edges in the resulting planar graph.

Original entry on oeis.org

3, 21, 8, 132, 92, 20, 429, 596, 290, 42, 1272, 1936, 2215, 708, 91, 2826, 6020, 7405, 4020, 1575, 136, 5640, 11088, 21150, 15120, 10962, 2632, 288, 10461, 26260, 43490, 38544, 35812, 17728, 5148, 390, 17094, 42144, 88230, 83136, 96257, 60672, 33291, 7800, 715

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Nov 14 2023

See A367322, A367323 and the cross references for images of the n-gons.

			The table begins:
3, 21, 132, 429, 1272, 2826, 5640, 10461, 17094, 26847, 41046, 61041, 84051, ...
8, 92, 596, 1936, 6020, 11088, 26260, 42144, 72296, 107832, 183340, 222940, ...
20, 290, 2215, 7405, 21150, 43490, 88230, 151135, 250825, 384360, 578840, ...
42, 708, 4020, 15120, 38544, 83136, 169686, 294678, 475500, 746340, 1140624, ...
91, 1575, 10962, 35812, 96257, 201054, 389991, 668458, 1096508, 1675835, ...
136, 2632, 17728, 60672, 163776, 341920, 673112, 1155144, 1892528, 2905088, ...
288, 5148, 33291, 108252, 283464, 591723, 1133928, 1941786, 3166605, 4837824, ...
390, 7800, 48870, 164470, 430840, 900890, 1735800, 2982660, 4849740, 7438490, ...
715, 12793, 79134, 255552, 660033, 1376870, 2619287, 4482654, 7284904, ...
756, 16512, 99348, 346140, 912960, 1894920, 3685056, 6313164, 10261200, ...
1508, 26806, 160641, 516932, 1322802, 2757339, 5221996, 8932664, 14483183, ...
1722, 35546, 210658, 696682, 1773828, 3718400, 7030464, 12067720, 19517596, ...
2835, 49995, 292590, 939720, 2388825, 4976130, 9394815, 16064970, 26003640, ...
3088, 63456, 370784, 1217664, 3081472, 6455872, 12162640, 20861328, 33700320, ...
4896, 85680, 493017, 1579436, 3995102, 8318525, 15667336, 26783636, ...
4320, 99036, 593784, 1958922, 4978872, 10395450, 19644408, ...
7923, 137693, 781470, 2499792, 6298633, 13109658, 24645983, ...
8360, 167160, 941940, 3068280, 7705420, 16112480, 30238400, ...
12180, 210378, 1180683, 3772692, 9476418, 19717089, ...
12782, 252296, 1400674, 4547884, 11375584, 23776236, ...
17963, 308591, 1716306, 5478232, 13725457, 28550084, ...
16344, 350448, 1981416, 6460080, 16185624, ...
25600, 437700, 2415825, 7704700, 19262750, ...
.
.
.

Cf. A367322 (vertices), A367323 (regions), A274586 (1st row), A331448 (2nd row), A329710 (3rd row), A330845 (4th row), A333112 (5th row), A333110 (6th row), A332429 (7th row), A332419 (8th row), A135565 (1st column).

T(n,k) = A367322(n,k) + A367323(n,k) - 1 (Euler).

cp's OEIS Frontend

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.

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A331449 a(0) = 1 by convention; for n>0, a(n) is the number of points in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

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A331454 Triangle read by rows: T(n,m) (n >= m >= 1) = number of line segments formed by drawing the lines connecting any two of the 2*(m+n) perimeter points of an m X n grid of squares.

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A334690 a(n) is the number of points in the interior of the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

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A367276 Place n equally spaced points on each side of a square, and join each of these points by a chord to the 3*n new points on the other three sides: sequence gives number of vertices in the resulting planar graph.

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A345650 Number of polygon edges formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

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A367122 Place n points in general position on each side of a square, and join every pair of the 4*n+4 boundary points by a chord; sequence gives number of edges in the resulting planar graph.

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A358409 Number of edges formed in a square by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on each of the two adjacent edges of the square.

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A367324 Table read by antidiagonals: Place k equally spaced points on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives the number of edges in the resulting planar graph.

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