A331449 -id:A331449 - 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

A331453 Triangle read by rows: T(n,m) (n >= m >= 1) = number of vertices 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

5, 13, 37, 35, 99, 257, 75, 213, 421, 817, 159, 401, 881, 1489, 2757, 275, 657, 1305, 2143, 3555, 4825, 477, 1085, 2131, 3431, 5821, 7663, 12293, 755, 1619, 2941, 4817, 7477, 9913, 15037, 19241, 1163, 2327, 4369, 6495, 10393, 13647, 20425, 24651, 33549, 1659, 3257, 5603, 8637, 13689, 16953, 25125, 30779, 39857, 49577

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 vertices in the resulting diagram, and A331452(m,n) and A331454(m,n) give the number of regions and the number of line segments respectively.

For illustrations see the links in A331452.

			Triangle begins:
5,
13, 37,
35, 99, 257,
75, 213, 421, 817,
159, 401, 881, 1489, 2757,
275, 657, 1305, 2143, 3555, 4825,
477, 1085, 2131, 3431, 5821, 7663, 12293,
755, 1619, 2941, 4817, 7477, 9913, 15037, 19241,
1163, 2327, 4369, 6495, 10393, 13647, 20425, 24651, 33549,
...

Lars Blomberg, Table of n, a(n) for n = 1..703 (the first 37 rows)
Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
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.
N. J. A. Sloane, Conant's Gasket, Recamán Variations, the Enots Wolley Sequence, and Stained Glass Windows, Experimental Math Seminar, Rutgers University, Sep 10 2020 (video of Zoom talk)

The main diagonal is A331449.
The first two columns are A331755 and A331763.
Cf. A288187, A331452, A331454.

A331448 a(0) = 0 by convention; for n>0, a(n) is the number of edges in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

Original entry on oeis.org

0, 8, 92, 596, 1936, 6020, 11088, 26260, 42144, 72296, 107832, 183340, 222940, 371100, 466936, 609916, 804504, 1139632, 1288536, 1813288, 2012676, 2536572, 3142008, 3997580, 4230340, 5430444, 6331892, 7360512, 8262568, 10367804

Offset: 0

N. J. A. Sloane, Jan 23 2020, based on a comment from Michael De Vlieger in A255011

nonn

By Euler's formula, a(n) = A331449(n) + A255011(n) - 1.

Zhao Hui Du, Table of n, a(n) for n = 0..136 (terms 0..52 from Lars Blomberg).

Cf. A255011, A331449.

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

a(11)-a(29) from Giovanni Resta, Jan 28 2020

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.

A334691 Irregular triangle read by rows: T(n,k) (n >= 1, 2 <= k <= 2*n) = number of interior vertices in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter) where k lines meet.

Original entry on oeis.org

1, 20, 8, 1, 204, 32, 8, 0, 1, 616, 152, 20, 8, 4, 0, 1, 2428, 252, 36, 16, 4, 0, 0, 0, 1, 3968, 572, 156, 72, 16, 8, 4, 0, 4, 0, 1, 11164, 900, 120, 52, 16, 8, 4, 0, 0, 0, 0, 0, 1, 16884, 1712, 396, 132, 40, 20, 8, 8, 8, 0, 0, 0, 0, 0, 1, 30116, 2536, 600, 140, 60, 24, 8, 20, 8, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 18 2020

No formula is known.

			Triangle begins:
1;
20,8,1;
204,32,8,0,1;
616,152,20,8,4,0,1;
2428,252,36,16,4,0,0,0,1;
3968,572,156,72,16,8,4,0,4,0,1;
11164,900,120,52,16,8,4,0,0,0,0,0,1;
16884,1712,396,132,40,20,8,8,8,0,0,0,0,0,1;
30116,2536,600,140,60,24,8,20,8,0,0,0,0,0,0,0,1;
43988,4056,948,312,84,56,52,20,,8,0,0,4,0,0,0,0,0,1;
82016,4660,580,228,48,84,4,4,4,8,4,0,0,0,0,0,0,0,0,0,1;
90088,8504,1840,780,424,128,68,32,32,0,0,8,24,0,0,0,4,0,0,0,0,0,1;
168360,8284,1056,396,128,100,52,12,4,4,4,8,4,0,0,0,0,0,0,0,0,0,0,0,1;
202332,13144,2980,924,256,144,140,60,44,4,0,8,8,8,0,0,4,0,0,0,0,0,0,0,0,0,1;
...

Scott R. Shannon, Data for triangles A334691 and A334699
Scott R. Shannon, Colored illustration for n = 2
Scott R. Shannon, Illustration for n=3 showing interior vertices color-coded according to multiplicity.

Cf. A255011, A331449, A334690 (row sums), A334692 (column k=2), A334693 (k=3), A334694-A334699.

A334698 a(n) is the total number of points (both boundary and interior points) in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter), where the interior points are counted with multiplicity.

Original entry on oeis.org

5, 58, 375, 1376, 3685, 8130, 15743, 27760, 45621, 70970, 105655, 151728, 211445, 287266, 381855, 498080, 639013, 807930, 1008311, 1243840, 1518405, 1836098, 2201215, 2618256, 3091925, 3627130, 4228983, 4902800, 5654101, 6488610, 7412255, 8431168, 9551685, 10780346, 12123895, 13589280, 15183653, 16914370, 18788991

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 18 2020

An equivalent definition: Place n-1 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 vertices in the resulting planar graph. "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. - Scott R. Shannon and N. J. A. Sloane, Nov 05 2023
Equivalently, this is A334697(n) + 4*n.
This is an upper bound on A331449.

Colin Barker, Table of n, a(n) for n = 1..1000
Scott R. Shannon, Colored illustration for n = 2
Scott R. Shannon, Illustration for n=3 showing interior vertices color-coded according to multiplicity.
Scott R. Shannon, "General position" image for n = 1.
Scott R. Shannon, "General position" image for n = 2.
Scott R. Shannon, "General position" image for n = 3.
Scott R. Shannon, "General position" image for n = 4.
Scott R. Shannon, "General position" image for n = 5.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A255011, A331449, A334690-A334698.

For the "general position" version, see also A367121 (regions), A367122 (edges), and A367117.

A334698[n_]:=n(17n^3-30n^2+19n+4)/2;Array[A334698,50] (* or *)
LinearRecurrence[{5,-10,10,-5,1},{5,58,375,1376,3685},50] (* Paolo Xausa, Nov 14 2023 *)

Vec(x*(5 + 33*x + 135*x^2 + 31*x^3) / (1 - x)^5 + O(x^40)) \\ Colin Barker, May 31 2020

Theorem: a(n) = n*(17*n^3-30*n^2+19*n+4)/2.
From Colin Barker, May 27 2020: (Start)
G.f.: x*(5 + 33*x + 135*x^2 + 31*x^3) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
(End)
a(n+1) = A367122(n) - A367121(n) + 1 by Euler's formula.

Edited by N. J. A. Sloane, Nov 13 2023

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).

A334699 Triangle read by rows: T(n,k) (n >= 1, 3 <= k <= n+2) = number of k-sided cells in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

Original entry on oeis.org

4, 48, 8, 204, 112, 24, 680, 384, 56, 0, 1660, 1072, 448, 76, 8, 3416, 2392, 400, 56, 0, 0, 6348, 5236, 1840, 504, 40, 0, 0, 11048, 8624, 2800, 408, 24, 0, 0, 0, 17708, 14976, 4960, 1004, 96, 4, 0, 0, 0, 27520, 22200, 7104, 1288, 136, 8, 0, 0, 0, 0, 40388, 35832, 15080, 3808, 488, 60, 0, 0, 0, 0, 0

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 18 2020

No formula is known.

For further illustrations see A331452.

			Triangle begins:
      4;
     48,     8;
    204,   112,    24;
    680,   384,    56,    0;
   1660,  1072,   448,   76,    8;
   3416,  2392,   400,   56,    0,   0;
   6348,  5236,  1840,  504,   40,   0,  0;
  11048,  8624,  2800,  408,   24,   0,  0, 0;
  17708, 14976,  4960, 1004,   96,   4,  0, 0, 0;
  27520, 22200,  7104, 1288,  136,   8,  0, 0, 0, 0;
  40388, 35832, 15080, 3808,  488,  60,  0, 0, 0, 0, 0;
  57360, 47160, 13856, 2272,  296,  16,  0, 0, 0, 0, 0, 0;
  79108, 73572, 30800, 7788, 1152, 200, 16, 0, 0, 0, 0, 0, 0;
...

Scott R. Shannon, Data for triangles A334691 and A334699
Scott R. Shannon, Colored illustration for n = 2
Scott R. Shannon, Image showing the k-sided cells for n = 10.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 20.

Cf. A255011 (row sums), A331452, A331449, A334690, A334691, A334695, A334696, A334700.

a(29) and beyond by Scott R. Shannon, Feb 18 2021

A334697 a(n) is the number of interior points in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter), counted with multiplicity.

Original entry on oeis.org

1, 50, 363, 1360, 3665, 8106, 15715, 27728, 45585, 70930, 105611, 151680, 211393, 287210, 381795, 498016, 638945, 807858, 1008235, 1243760, 1518321, 1836010, 2201123, 2618160, 3091825, 3627026, 4228875, 4902688, 5653985, 6488490, 7412131, 8431040, 9551553, 10780210, 12123755, 13589136, 15183505, 16914218, 18788835

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 18 2020

			Scott Shannon's illustration for n=2 shows 29 interior intersection points, of which 20 are simple intersections, 8 are triple intersections, and one (the central point) is a 4-fold intersection. A point where d lines meet is equivalent to C(d,2) simple points. So a(2) = 20*1 + 8*3 + 1*6 = 50.

Colin Barker, Table of n, a(n) for n = 1..1000
Scott R. Shannon, Colored illustration for n = 2
Scott R. Shannon, Illustration for n=3 showing interior vertices color-coded according to multiplicity.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A255011, A331449, A334690-A334698.

Vec(x*(1 + 45*x + 123*x^2 + 35*x^3) / (1 - x)^5 + O(x^30)) \\ Colin Barker, May 31 2020

Theorem: a(n) = n*(17*n^3-30*n^2+19*n-4)/2.
From Colin Barker, May 27 2020: (Start)
G.f.: x*(1 + 45*x + 123*x^2 + 35*x^3) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
(End)

A334700 Triangle read by rows: T(n,k) (n >= 1, 3 <= k <= n+2) = one-quarter of the number of k-sided cells in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

Original entry on oeis.org

1, 12, 2, 51, 28, 6, 170, 96, 14, 0, 415, 268, 112, 19, 2, 854, 598, 100, 14, 0, 0, 1587, 1309, 460, 126, 10, 0, 0

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 18 2020

This is the triangle in A334699 with all entries divided by 4.
No formula is known.
For illustrations see A331452.

			Triangle begins:
1,
12, 2,
51, 28, 6,
170, 96, 14, 0,
415, 268, 112, 19, 2,
854, 598, 100, 14, 0, 0,
1587, 1309, 460, 126, 10, 0, 0
...

Scott R. Shannon, Data for triangles A334691 and A334699
Scott R. Shannon, Colored illustration for n = 2

Cf. A255011, A331452, A331449, A334690, A334691, A334695, A334696, A334699.

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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A331453 Triangle read by rows: T(n,m) (n >= m >= 1) = number of vertices 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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A331448 a(0) = 0 by convention; for n>0, a(n) is the number of edges in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

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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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A334691 Irregular triangle read by rows: T(n,k) (n >= 1, 2 <= k <= 2*n) = number of interior vertices in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter) where k lines meet.

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A334698 a(n) is the total number of points (both boundary and interior points) in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter), where the interior points are counted with multiplicity.

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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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A334699 Triangle read by rows: T(n,k) (n >= 1, 3 <= k <= n+2) = number of k-sided cells in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

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A334697 a(n) is the number of interior points in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter), counted with multiplicity.

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