A331452 -id:A331452 - cp's OEIS frontend

A333281 Column 2 of triangle in A288180.

13, 37, 121, 265, 587, 1019, 1797, 2823, 4369, 6257, 9001, 12289, 16775, 21905, 28383, 35901, 45463, 56119, 69351, 84167, 101687, 120869, 143777, 168873, 198191, 229771, 266015, 305379, 350673, 399035, 454243, 513619, 579787, 649899, 727927, 810907, 903581

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 15 2020

nonn

For the graphs defined in A331452 and A288187 only the counts for graphs that are one square wide have formulas for regions, edges, and vertices (see A306302, A331757, A331755). For width 2 there are six such sequences (A331766, A331765, A331763; A333279, A333280, A333281). It would be nice to have a formula for any one of them.

See A333279 for illustrations.

Lars Blomberg, Table of n, a(n) for n = 1..200
Hugo Pfoertner, Illustrations of Chamber Complexes up to 5 X 5.

Cf. A331452, A288187; A331766, A331765, A331763; A333279, A333280, A333281.

a(10) and beyond from Lars Blomberg, May 23 2021

A333282 Triangle read by rows: T(m,n) (m >= n >= 1) = number of regions formed by drawing the line segments connecting any two of the (m+1) X (n+1) lattice points in an m X n grid of squares and extending them to the boundary of the grid.

Original entry on oeis.org

4, 16, 56, 46, 192, 624, 104, 428, 1416, 3288, 214, 942, 3178, 7520, 16912, 380, 1672, 5612, 13188, 29588, 51864, 648, 2940, 9926, 23368, 52368, 92518, 164692, 1028, 4624, 15732, 37184, 83628, 148292, 263910, 422792, 1562, 7160, 24310, 57590, 130034, 230856, 410402, 658080, 1023416

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 16 2020

Triangle gives number of nodes in graph LC(m,n) in the notation of Blomberg-Shannon-Sloane (2020).

If we only joined pairs of the 2(m+n) boundary points, we would get A331452. If we did not extend the lines to the boundary of the grid, we would get A288187. (One of the links below shows the difference between the three definitions in the case m=3, n=2.)

			Triangle begins:
4,
16, 56,
46, 192, 624,
104, 428, 1416, 3288,
214, 942, 3178, 7520, 16912,
380, 1672, 5612, 13188, 29588, 51864,
648, 2940, 9926, 23368, 52368, 92518, 164692,
1028, 4624, 15732, 37184, 83628, 148292, 263910, 422792
1562, 7160, 24310, 57590, 130034, 230856, 410402, 658080, 1023416
2256, 10336, 35132, 83116, 187376, 331484, 588618, 942808, 1466056, 2101272

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
Seppo Mustonen, Statistical accuracy of geometric constructions, 2008.
Seppo Mustonen, Statistical accuracy of geometric constructions, 2008 [Local copy]
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points, 2009
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points, 2009 [Local copy]
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points, 2010
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points, 2010 [Local copy]
Scott R. Shannon, Colored illustration for T(3,2)
Scott R. Shannon, Colored illustration for T(3,2) (edge number coloring)
Scott R. Shannon, Colored illustration for T(3,3)
Scott R. Shannon, Colored illustration for T(3,3) (edge number coloring)
Scott R. Shannon, Colored illustration for T(4,4)
Scott R. Shannon, Colored illustration for T(4,4) (edge number coloring)
Scott R. Shannon, Colored illustration for T(5,5)
Scott R. Shannon, Colored illustration for T(5,5) (edge number coloring)
Scott R. Shannon, Colored illustration for T(6,3)
Scott R. Shannon, Colored illustration for T(6,3) (edge number coloring)
Scott R. Shannon, Colored illustration for T(7,4)
Scott R. Shannon, Colored illustration for T(7,4) (edge number coloring)
Scott R. Shannon, Colored illustration for T(8,2)
Scott R. Shannon, Colored illustration for T(8,2) (edge number coloring)
Scott R. Shannon, Numerical properties of these structures. Corrected version Mar 24 2020.
N. J. A. Sloane, Illustration of T(3,2) = 192. [Black lines correspond to A331452(3,2), black + red lines correspond to A288187(3,2), and black + red + blue lines to T(3,2)]
N. J. A. Sloane, Illustration of T(3,3) = 624 [Black lines correspond to A288187(3,3), and black + red lines to T(3,3)]

Cf. A288187, A331452, A333283 (edges), A333284 (vertices). Column 1 is A306302. Main diagonal is A333294.

More terms and corrections from Scott R. Shannon, Mar 21 2020

More terms from Scott R. Shannon, May 27 2021

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

A355839 Number of vertices 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

5, 25, 133, 357, 1013, 1637, 3761, 5561, 9313, 13065, 21689, 25357, 41553, 50157, 66005, 84897, 117793, 129841, 181717, 198857, 251189, 302293, 383161, 401073, 517193, 587041, 687765, 763425, 949869, 966249, 1234425, 1320913, 1512233, 1703657, 1912765, 2023569, 2475361, 2649813, 2934997

Offset: 1

Scott R. Shannon, Jul 18 2022

nonn

This sequence is similar to A355799 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 = 11.

Cf. A355838 (regions), A355840 (edges), A355841 (k-gons), A355799 (without corner vertices), A290131, A331452, A335678.

a(n) = A355840(n) - A355838(n) + 1 by Euler's formula.

A355949 Number of vertices formed in a square by straight line segments when connecting the four corner vertices to the points dividing the sides into n equal parts.

Original entry on oeis.org

5, 25, 81, 157, 301, 381, 665, 821, 1109, 1353, 1825, 1861, 2621, 2881, 3285, 3813, 4645, 4773, 5873, 5953, 6821, 7665, 8761, 8613, 10165, 10921, 11777, 12337, 14173, 13717, 16265, 16581, 17861, 19161, 20093, 20461, 23405, 24145, 25305, 25701, 28885, 28433, 31841, 32077, 33269, 35841, 38185

Offset: 1

Scott R. Shannon, Jul 21 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 = 11.

Cf. A108914 (regions), A355948 (edges), A355992 (k-gons), A355839, A331452, A335678.

a(n) = A355948(n) - A108914(n) + 1 by Euler's formula.

A329714 Irregular table read by rows: Take a heptagon with all diagonals drawn, as in A329713. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

Original entry on oeis.org

35, 7, 7, 0, 1, 504, 224, 112, 28, 2331, 1883, 1008, 273, 92, 7, 7658, 6314, 3416, 798, 182, 28, 18662, 17514, 8463, 2898, 714, 175, 28, 7, 0, 0, 0, 1, 40404, 35462, 18508, 5796, 1330, 266, 28

Offset: 1

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

See the links in A329713 for images of the heptagons.

			A heptagon with no other points along its edges, n = 1, contains 35 triangles, 7 quadrilaterals, 7 pentagons, 1 heptagon and no other n-gons, so the first row is [35,7,7,0,1]. A heptagon with 1 point dividing its edges, n = 2, contains 504 triangles, 224 quadrilaterals, 112 pentagons, 28 hexagons and no other n-gons, so the second row is [504,224,112,28].
Triangle begins:
35, 7, 7, 0, 1;
504, 224, 112, 28;
2331, 1883, 1008, 273, 92, 7;
7658, 6314, 3416, 798, 182, 28;
18662, 17514, 8463, 2898, 714, 175, 28, 7, 0, 0, 0, 1;
40404, 35462, 18508, 5796, 1330, 266, 28;
73248, 71596, 35777, 11669, 2654, 651, 70, 49;
The row sums are A329713.

Lars Blomberg, Table of n, a(n) for n = 1..251 (the first 27 rows)

Cf. A329713 (regions), A333112 (edges), A333113 (vertices), A331906, A007678, A092867, A331452.

A331455 Number of regions in a "cross" of width 3 and height n (see Comments for definition).

Original entry on oeis.org

64, 104, 176, 304, 492, 778, 1176, 1732, 2446, 3416, 4614, 6172, 8060, 10340, 13052, 16388, 20228, 24852, 30134, 36206, 43076, 51092, 60010, 70186, 81498, 94180, 108140, 123938, 141074, 160308, 181320, 204328, 229288, 256574, 285856, 318124, 352838, 390338

Offset: 2

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

nonn

This "cross" of height n consists of a vertical column of n >= 2 squares with two additional squares extending to the left and right of the second square. (See illustrations.)
There are n+2 squares in all. The number of vertices is 3*n+2.
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.

Lars Blomberg, Table of n, a(n) for n = 2..50
Scott R. Shannon, Illustration for cross of height 2.
Scott R. Shannon, Illustration for cross of height 3.
Scott R. Shannon, Illustration for cross of height 4.
Scott R. Shannon, Illustration for cross of height 5.
Scott R. Shannon, Illustration for cross of height 6.
Scott R. Shannon, Illustration for cross of height 9.
Scott R. Shannon, Illustration for cross of height 3 using random distance-based coloring.
Scott R. Shannon, Illustration for cross of height 4 using random distance-based coloring.
Scott R. Shannon, Illustration for cross of height 5 using random distance-based coloring.
Scott R. Shannon, Illustration for cross of height 6 using random distance-based coloring.
Scott R. Shannon, Illustration for cross of height 7 using random distance-based coloring.
Scott R. Shannon, Colored illustration for a different-shaped cross, with arms of lengths 2,2,4. (There are 21858 regions.)
N. J. A. Sloane, Illustration for cross of height 2.
N. J. A. Sloane, Illustration for cross of height 3. (One of the "arms" has been cropped by the scanner, but all four arms are the same.)
N. J. A. Sloane, Illustration for cross of height 4.
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. A330848 (n-gons), A330850 (vertices), A330851 (edges).
See A331456 for crosses in which the arms have equal length.
A331452 is a similar sequence for a rectangular region; A007678 for a polygonal region.

a(11) and beyond from Lars Blomberg, May 31 2020

A331907 Triangle read by rows: Take a pentagram with all diagonals drawn, as in A331906. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n+2.

Original entry on oeis.org

40, 0, 0, 590, 420, 80, 10, 2890, 3030, 1130, 230, 50, 9540, 10530, 4290, 980, 190, 10, 22730, 28390, 10960, 3200, 550, 80, 20, 47610, 57450, 23270, 6530, 1160, 160, 20, 0, 90080, 109160, 47430, 13430, 2460, 410, 40, 0, 0, 154840, 193480, 82330, 22410, 4620

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 31 2020

See the links in A331906 for images of the pentagrams.

			A pentagram with no other points along its edges, n = 1, contains 40 triangles and no other n-gons, so the first row is [40,0,0]. A pentagram with 1 point dividing its edges, n = 2, contains 590 triangles, 420 quadrilaterals, 80 pentagons and 10 hexagons, so the second row is [590,420,80,10].
Triangle begins:
40,0,0
590, 420, 80, 10
2890, 3030, 1130, 230, 50
9540, 10530, 4290, 980, 190, 10
22730, 28390, 10960, 3200, 550, 80, 20
47610, 57450, 23270, 6530, 1160, 160, 20, 0
The row sums are A331906.

Lars Blomberg, Table of n, a(n) for n = 1..250 (the first 20 rows)
Eric Weisstein's World of Mathematics, Pentagram.

Cf. A331906 (regions), A333117 (vertices), A333118 (edges), A007678, A092867, A331452.

a(34) and beyond from Lars Blomberg, May 06 2020

A331908 The number of regions inside a hexagram 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

168, 3588, 20424, 73860, 189468, 402216, 782808, 1385040, 2214108, 3423840, 5196312, 7218552, 10353432, 13823772, 18047124, 24083736, 31051152, 38334972, 48877440, 59201544, 72052956, 88004184, 106601088, 124009020

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 31 2020

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

Scott R. Shannon, Hexagram regions for n = 1.
Scott R. Shannon, Hexagram regions for n = 2.
Scott R. Shannon, Hexagram regions for n = 3.
Scott R. Shannon, Hexagram regions for n = 4.
Scott R. Shannon, Hexagram regions for n = 5.
Scott R. Shannon, Hexagram regions with random distance-based coloring for n = 1.
Scott R. Shannon, Hexagram regions with random distance-based coloring for n = 2.
Scott R. Shannon, Hexagram regions with random distance-based coloring for n = 3.
Eric Weisstein's World of Mathematics, Hexagram.

Cf. A331909 (n-gons), A333116 (vertices), A333049 (edges), A007678, A092867, A331452, A331906.

a(6)-a(24) from Lars Blomberg, May 10 2020

cp's OEIS Frontend

A333281 Column 2 of triangle in A288180.

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A333282 Triangle read by rows: T(m,n) (m >= n >= 1) = number of regions formed by drawing the line segments connecting any two of the (m+1) X (n+1) lattice points in an m X n grid of squares and extending them to the boundary of the grid.

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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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A355839 Number of vertices 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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A355949 Number of vertices formed in a square by straight line segments when connecting the four corner vertices to the points dividing the sides into n equal parts.

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A329714 Irregular table read by rows: Take a heptagon with all diagonals drawn, as in A329713. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

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A331455 Number of regions in a "cross" of width 3 and height n (see Comments for definition).

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A331907 Triangle read by rows: Take a pentagram with all diagonals drawn, as in A331906. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n+2.

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A331908 The number of regions inside a hexagram 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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