A331452 -id:A331452 - cp's OEIS frontend

A333288 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all visible rectangles; T(n,k) (1 <= k <= n) is the number of regions in the k-th rectangle.

4, 8, 8, 12, 22, 12, 16, 36, 36, 16, 20, 52, 70, 52, 20, 24, 66, 100, 100, 66, 24, 28, 82, 134, 160, 134, 82, 28, 32, 98, 166, 218, 218, 166, 98, 32, 36, 116, 198, 276, 310, 276, 198, 116, 36, 40, 134, 230, 328, 396, 396, 328, 230, 134, 40, 44, 154, 266, 386

Offset: 1

N. J. A. Sloane, Mar 20 2020

This was originally based on the data in Jinyuan Wang's A324042, and then extended by Lars Blomberg.
Since the cells are either triangles or quadrilaterals, this is the sum of the two arrays A333286 and A333287.
It would be nice to have a formula for these entries. It is easy to see that the first column is 4n for n>=1.

			Triangle begins:
   4;
   8,   8;
  12,  22,  12;
  16,  36,  36,  16;
  20,  52,  70,  52,  20;
  24,  66, 100, 100,  66,  24;
  28,  82, 134, 160, 134,  82,  28;
  ...

Lars Blomberg, Table of n, a(n) for n = 1..3240 (the first 80 rows)
Lars Blomberg, Scott R. Shannon and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.

Cf. A306302, A331452, A324042, A324043, A333286, A333287, A333288, A335056, A335074.

a(29) and beyond from Lars Blomberg, Apr 23 2020

A290865 a(n) = number of regions in the configuration A290447(n).

Original entry on oeis.org

0, 1, 3, 7, 15, 30, 56, 98, 161, 250, 370, 536, 748, 1027, 1379, 1807, 2320, 2954, 3702, 4604, 5652, 6852, 8239, 9858, 11683, 13748, 16086, 18700, 21604, 24887, 28471, 32491, 36907, 41751, 47080, 52876, 59105, 65965, 73440, 81521, 90176

Offset: 1

David Applegate, Aug 12 2017

nonn

			With 3 points, there are 3 semicircles above the baseline, which bound a(3) = 3 regions. With 4 points, there are 6 semicircles, defining 7 regions (use the Halser webpage with n = 3 and 4). - _N. J. A. Sloane_, Aug 12 2017

David Applegate, Table of n, a(n) for n = 1..100
M. F. Hasler, Interactive web page for drawing the illustration for a(n).
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.

Cf. A290447, A290866, A290867, A332723 (number of regions with k edges).

A331456 Number of regions in an equal-armed cross with arms of length n (see Comments for definition).

Original entry on oeis.org

4, 104, 568, 1900, 4808, 10180, 19180, 33132, 53628, 82432, 121448, 172948, 239356, 323168, 427272, 554892, 709476, 893772, 1111588, 1367292, 1664604, 2008240, 2402560, 2852532, 3363280, 3938712, 4585568, 5308720, 6112736, 7006068, 7994412, 9084788, 10281812

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Jan 28 2020

nonn

This cross of height n consists of a central square with 4 arms of length n.
There are 4n+1 squares in all. The number of vertices is 8n+4.
Now join every pair of vertices by a line segment, provided the line does not extend beyond the boundary of the cross. The sequence gives the number of regions in the resulting figure.
See A337641 for information about these regions, their numbers of sides, their coordinates, and for further illustrations. - N. J. A. Sloane, Sep 17 2020

Lars Blomberg, Table of n, a(n) for n = 0..48
Scott R. Shannon, Colored illustration for a(0).
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, Colored illustration for a(9).
Scott R. Shannon, Colored illustration for a(1) classifying nodes and cells.
Scott R. Shannon, Colored illustration for a(2) classifying nodes and cells.
Scott R. Shannon, Colored illustration for a(3) classifying nodes and cells.
Scott R. Shannon, Colored illustration for a(4) classifying nodes and cells.
Scott R. Shannon, Colored illustration for a(5) classifying nodes and cells.
Scott R. Shannon, Colored illustration for a(6) classifying nodes and cells.
N. J. A. Sloane, Illustration for a(1). (One of the "arms" has been cropped by the scanner, but all four arms are the same.)
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.

Cf. A333035 (vertices), A333036 (edges), A333037 (n-gons), A337641.
See A331455 for a different family of crosses.
A331452 is a similar sequence for a rectangular region; A007678 for a polygonal region.
Cf. A331458.

a(11) and beyond from Lars Blomberg, May 30 2020

A331457 Triangle read by rows: T(n,k) = number of regions in a "frame" of size n X k (see Comments for definition).

Original entry on oeis.org

4, 16, 56, 46, 142, 208, 104, 296, 348, 496, 214, 544, 592, 752, 1016, 380, 892, 948, 1120, 1396, 1784, 648, 1436, 1508, 1692, 1980, 2380, 2984, 1028, 2136, 2292, 2488, 2788, 3200, 3816, 4656, 1562, 3066, 3384, 3592, 3904, 4328, 4956, 5808, 6968, 2256, 4272, 4796, 5016, 5340, 5776, 6416, 7280, 8452, 9944

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 29 2020

A "frame" of size n X k is formed from a grid of (n+1) X (k+1) points with the central grid of (n-3) X (k-3) points removed. If n or k is less than 3 then no points are removed, and T(n,k) = A331452(n,k). From now on we assume both n and k are >= 3.
The resulting array has an outer perimeter with 2*(n+k) points and an inner perimeter with 2*(n+k)-8 points, for a total of 4*(n+k)-8 perimeter points. The frame itself is the strip of width 1 between the inner and outer perimeters.
Now join every pair of perimeter points, both inner and outer, by a line segment, provided the line remains inside the frame. The sequence gives the number of regions in the resulting figure.
See A331776 for additional illustrations for the diagonal entries.
There is a crucial difference between frames of size nX2 and size nXk with k = 1 or k >= 3. If k != 2, all regions are either triangles or quadrilaterals, but for k=2 regions with larger numbers of sides can appear. Remember also that for k <= 2, the "frame" has no hole, and the graph has genus 0, whereas for k >= 3 there is a nontrivial hole and the graph has genus 1.

			Triangle begins:
4,
16,56,
46,142,208,
104,296,348,496,
214,544,592,752,1016
380,892,948,1120,1396,1784
648,1436,1508,1692,1980,2380,2984
1028,2136,2292,2488,2788,3200,3816,4656
1562,3066,3384,3592,3904,4328,4956,5808,6968
2256,4272,4796,5016,5340,5776,6416,7280,8452,9944

Scott R. Shannon, Colored illustration for T(1,1) = 4.
Scott R. Shannon, Colored illustration for T(2,2) = 56.
Scott R. Shannon, Colored illustration for T(3,3) = 208.
Scott R. Shannon, Colored illustration for T(4,4) = 496.
Scott R. Shannon, Colored illustration for T(5,5) = 1016.
Scott R. Shannon, Colored illustration for T(6,6) = 1784.
Scott R. Shannon, Colored illustration for T(7,4) = 1692.
Scott R. Shannon, Colored illustration for T(10,6) = 5776.
N. J. A. Sloane, Illustration for T(3,3) = 208.

Cf. A332599 (triangle giving numbers of vertices) and A332600 (edges).
Cf. also A331452.
The first column is A306302, the main diagonal is A331776.

Column 1 is A306302, for which there is an explicit formula.
Column 2 is A331766, for which no formula is known.
For n >= k >= 3, T(n,k) = A332610(n,k) + A332611(n,k), both of which have explicit formulas.

More terms from Scott R. Shannon, Mar 05 2020

a(8) corrected by Giovanni Resta, May 22 2025

A333075 The number of regions inside an octagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Original entry on oeis.org

80, 1488, 9312, 31552, 83432, 174816, 339816, 584176, 953416, 1463936, 2173976, 3074784, 4294080, 5790816, 7664880, 9952944, 12757088, 16036096, 20013696, 24577760, 29973528, 36161472, 43314312, 51334672

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 07 2020

The terms are from numeric computation - no formula for a(n) is currently known.

Scott R. Shannon, Octagon regions for n = 1.
Scott R. Shannon, Octagon regions for n = 2.
Scott R. Shannon, Octagon regions for n = 3.
Scott R. Shannon, Octagon regions for n = 4.
Scott R. Shannon, Octagon regions using random distance-based coloring for n = 2.
Scott R. Shannon, Octagon regions using random distance-based coloring for n = 3.
Scott R. Shannon, Octagon regions using random distance-based coloring for n = 4.
Wikipedia, Octagon.

Cf. A333076 (n-gons), A333109 (vertices), A333110 (edges), A007678, A092867, A331452, A331931.

a(7)-a(24) from Lars Blomberg, May 14 2020

A331911 Triangle read by rows: Take an equilateral triangle with all diagonals drawn, as in A092867. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n+2 and where n is the number of equal parts each side is divided into.

Original entry on oeis.org

1, 12, 0, 48, 24, 3, 162, 90, 0, 0, 378, 306, 15, 16, 0, 774, 696, 84, 18, 0, 0, 1470, 1383, 219, 37, 0, 0, 0, 2604, 2382, 600, 78, 6, 6, 0, 0, 4224, 4089, 771, 177, 24, 6, 0, 0, 0, 6624, 6186, 1470, 234, 42, 0, 0, 0, 0, 0, 9738, 9486, 2307, 498, 48, 0, 0, 3, 0, 1, 0, 14010, 13548, 3984, 816, 144, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 01 2020

			An equilateral triangle with no other point along its edges, n = 1, contains 1 triangle so the first row is [1]. An equilateral triangle with 1 point dividing its edges, n = 2, contains 12 triangles and no other n-gons, so the second row is [12,0]. An equilateral triangle with 2 points dividing its edges, n = 3, contains 48 triangles, 24 quadrilaterals and 3 pentagons, so the third row is [48,24,3].
Triangle begins:
1
12,0
48,24,3
162,90,0,0
378,306,15,16,0
774,696,84,18,0,0
1470,1383,219,37,0,0,0
2604,2382,600,78,6,6,0,0
4224,4089,771,177,24,6,0,0,0
6624,6186,1470,234,42,0,0,0,0,0
9738,9486,2307,498,48,0,0,3,0,1,0
14010,13548,3984,816,144,0,0,0,0,0,0,0
19248,19224,5007,1102,156,18,0,0,0,0,0,0,0
26208,26142,8634,1668,192,24,0,0,0,0,0,0,0,0
The row sums are A092867.

Hugo Pfoertner, Intersections of diagonals in polygons of triangular shape.
Scott R. Shannon, Triangle regions for n = 2.
Scott R. Shannon, Triangle regions for n = 3.
Scott R. Shannon, Triangle regions for n = 4.
Scott R. Shannon, Triangle regions for n = 5.
Scott R. Shannon, Triangle regions for n = 6.
Scott R. Shannon, Triangle regions for n = 7.
Scott R. Shannon, Triangle regions for n = 8.
Scott R. Shannon, Triangle regions for n = 9.
Scott R. Shannon, Triangle regions for n = 10.
Scott R. Shannon, Triangle regions for n = 11.
Scott R. Shannon, Triangle regions for n = 12.
Scott R. Shannon, Triangle regions for n = 13.
Scott R. Shannon, Triangle regions for n = 14.
Scott R. Shannon, Triangle regions for n = 9, random distance-based coloring.
Scott R. Shannon, Triangle regions for n = 12, random distance-based coloring

Cf. A092867, A092866, A092866, A274586, A331451, A331452, A331907, A331909

A355799 Number of vertices 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 the edge on the opposite side of the square.

Original entry on oeis.org

4, 9, 25, 93, 277, 597, 1405, 2421, 4357, 6661, 11261, 14593, 23625, 30121, 41453, 54477, 75985, 87677, 122433, 139461, 177965, 216017, 275733, 298805, 383497, 439909, 522473, 588597, 729501, 763149, 963573, 1045701, 1204481, 1361789, 1546309, 1657125, 2009113, 2166617, 2418733, 2602789

Offset: 1

Scott R. Shannon, Jul 17 2022

nonn

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 = 6.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 16.

Cf. A355798 (regions), A355800 (edges), A355801 (k-gons), A255011 (all vertices), A290131, A331452, A335678.

a(n) = A355800(n) - A355798(n) + 1 by Euler's formula.

A355838 Number of regions formed in a square by straight line segments when connecting the n+1 points along each edge that divide it into n equal parts to the n+1 points on the edge on the opposite side of the square.

Original entry on oeis.org

4, 40, 184, 496, 1240, 2144, 4380, 6720, 10860, 15528, 24300, 30152, 46036, 57496, 75056, 96416, 129052, 148512, 198392, 225240, 279576, 336272, 415988, 453376, 565052, 648008, 754808, 848664, 1026040, 1085536, 1331532, 1452704, 1652684, 1862600, 2084888, 2247568, 2662092, 2887944, 3193744

Offset: 1

Scott R. Shannon, Jul 18 2022

nonn

This sequence is similar to A355798 but here the corner vertices of the square are also connected to points on the opposite edge.

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 = 6.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 11.
Scott R. Shannon, Image for n = 16.

Cf. A355839 (vertices), A355840 (edges), A355841 (k-gons), A355798 (without corner vertices), A290131, A331452, A335678.

a(n) = A355840(n) - A355839(n) + 1 by Euler's formula.

A329985 a(1) = 1 and for n > 0, a(n+1) = a(k) - a(n) where k is the number of terms equal to a(n) among the first n terms.

Original entry on oeis.org

1, 0, 1, -1, 2, -1, 1, 0, 0, 1, -2, 3, -2, 2, -2, 3, -3, 4, -3, 3, -2, 1, 1, -2, 4, -4, 5, -4, 4, -3, 4, -5, 6, -5, 5, -5, 6, -6, 7, -6, 6, -5, 4, -2, 1, 0, -1, 2, -1, 0, 2, -3, 2, 0, -1, 3, -4, 5, -4, 3, -1, 0, 1, -1, 2, -3, 5, -6, 7, -7, 8, -7, 7, -6, 5, -3

Offset: 1

Rémy Sigrist, Nov 26 2019

In other words, for n > 0, a(n+1) = a(o(n)) - a(n) where o is the ordinal transform of the sequence.

The sequence has interesting graphical features (see plot in Links section).

			The first terms, alongside their ordinal transform, are:
  n   a(n)  o(n)
  --  ----  ----
   1     1     1
   2     0     1
   3     1     2
   4    -1     1
   5     2     1
   6    -1     2
   7     1     3
   8     0     2
   9     0     3
  10     1     4

Rémy Sigrist, Table of n, a(n) for n = 1..25000
Rémy Sigrist, Density plot of the first 2^22 terms
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.

o(n) is A330334.

See A329981 for similar sequences.

A={1};For[n=2,n<=76,n++,A=Append[A,Part[A,Count[Table[Part[A,i],{i,1,n-1}],Part[A,n-1]]]-Part[A,n-1]]];A (* Joshua Oliver, Nov 26 2019 *)
Nest[Append[#, #[[Count[#, #[[-1]] ] ]] - #[[-1]]] &, {1}, 75] (* Michael De Vlieger, Dec 01 2019 *)

for (n=1, #(a=vector(76)), print1 (a[n]=if (n==1, 1, a[sum(k=1, n-1, a[k]==a[n-1])]-a[n-1])", "))

A355800 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 the edge on the opposite side of the square.

Original entry on oeis.org

4, 12, 48, 196, 592, 1308, 2992, 5236, 9296, 14332, 23704, 31432, 49592, 64208, 87712, 115524, 158776, 186660, 255464, 295532, 374200, 455064, 574024, 632836, 800568, 923764, 1092672, 1238412, 1515912, 1613148, 2001200, 2191124, 2516016, 2847668, 3223968, 3485484, 4167304, 4523992, 5042336

Offset: 1

Scott R. Shannon, Jul 17 2022

nonn

See A355798 for images of the squares.

Cf. A355798 (regions), A355799 (vertices), A355801 (k-gons), A255011 (all vertices), A290131, A331452, A335678.

a(n) = A355798(n) + A355799(n) - 1 by Euler's formula.

cp's OEIS Frontend

A333288 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all visible rectangles; T(n,k) (1 <= k <= n) is the number of regions in the k-th rectangle.

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A290865 a(n) = number of regions in the configuration A290447(n).

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A331456 Number of regions in an equal-armed cross with arms of length n (see Comments for definition).

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A331457 Triangle read by rows: T(n,k) = number of regions in a "frame" of size n X k (see Comments for definition).

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A333075 The number of regions inside an octagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

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A331911 Triangle read by rows: Take an equilateral triangle with all diagonals drawn, as in A092867. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n+2 and where n is the number of equal parts each side is divided into.

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A355799 Number of vertices 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 the edge on the opposite side of the square.

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A355838 Number of regions formed in a square by straight line segments when connecting the n+1 points along each edge that divide it into n equal parts to the n+1 points on the edge on the opposite side of the square.

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A329985 a(1) = 1 and for n > 0, a(n+1) = a(k) - a(n) where k is the number of terms equal to a(n) among the first n terms.

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A355800 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 the edge on the opposite side of the square.

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