A332599 -id:A332599 - cp's OEIS frontend

A333113 The number of vertices inside a heptagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

42, 708, 5369, 17417, 47796, 99261, 194278, 331955, 546805, 833946, 1245314, 1762265, 2461837, 3311680, 4402405, 5700598, 7322231, 9200878, 11494161, 14108123, 17224438, 20752264, 24894009, 29506128, 34854099, 40780391, 47552050

Offset: 1

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

See the links in A329713 for images of the heptagons.

Cf. A329713 (regions), A329714 (n-gons), A333112 (edges), A330846, A092866, A332599, A007569.

a(8)-a(27) from Lars Blomberg, May 13 2020

A332608 Number of pentagons in the graph formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of a 3 X (n+1) rectangular grid of points (or equally, a 2 X n grid of squares).

Original entry on oeis.org

0, 0, 4, 12, 24, 28, 80, 128, 112, 200, 236, 356, 472, 652, 656, 940, 1040, 1300, 1600, 1948, 2048, 2588, 2856, 3260, 3716, 4492, 4572, 5324, 5904, 6508, 7200, 8144, 8664, 10296, 10548, 11664, 12580, 13860, 14596, 15980, 17312, 18516, 19692, 22152, 22912

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 12 2020

nonn

See A331452 (the illustrations for T(n,2)) for pictures of these graphs.

Lars Blomberg, Table of n, a(n) for n = 1..100
Scott R. Shannon, Data specifically for nX2 (or 2Xn) rectangles

Cf. A331452, A331453, A331454, A331763, A331765, A331766, A332599, A332600, A331457, A332606, A332607, A332609.

a(21) and beyond from Lars Blomberg, Apr 28 2020

A333116 The number of vertices 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

109, 2761, 16387, 63943, 167071, 357919, 711895, 1283419, 2040187, 3173851, 4909351, 6730795, 9868711, 13101883, 16984963, 23055523, 29896135, 36496711, 47223703, 56703265, 68999605, 84927301, 103692535, 119208667

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 08 2020

See the links in A331908 for images of the hexagrams.

Index entries for sequences related to stained glass windows

Cf. A331908 (regions), A331909 (n-gons), A333049 (edges), A092866, A332599, A007569.

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

A333117 The number of vertices inside a pentagram 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

26, 866, 6771, 24221, 64306, 132701, 259761, 453016, 734131, 1134081, 1673056, 2384606, 3326391, 4478286, 5941196, 7710796, 9901136, 12407581, 15497721, 19088991, 23256266, 28021386, 33537586, 39846196, 47092241, 55136771, 64103776, 74213991, 85642556, 98039461

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 08 2020

nonn

See the links in A331906 for images of the pentagrams.

Cf. A331906 (regions), A331907 (n-gons), A333118 (edges), A092866, A332599, A007569.

a(7)-a(30) from Lars Blomberg, May 06 2020

A332609 Maximum number of edges in any cell in the graph formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of a 3 X (n+1) rectangular grid of points (or equally, a 2 X n grid of squares).

Original entry on oeis.org

4, 4, 5, 5, 5, 6, 5, 6, 8, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 12 2020

nonn

See A331452 (the illustrations for T(n,2)) for pictures of these graphs.

Scott R. Shannon, Data specifically for nX2 (or 2Xn) rectangles

Cf. A331452, A331453, A331454, A331763, A331765, A331766, A332599, A332600, A331457, A332606, A332607, A332608.

a(21)-a(87) from Lars Blomberg, Apr 28 2020

A333136 The number of vertices formed on a triangle with leg lengths equal to the Pythagorean triples by the straight line segments mutually connecting all vertices and all points that divide the sides into unit length parts.

Original entry on oeis.org

230, 5138, 13181, 29277, 48107, 100003, 173261, 256910, 247940, 541752, 554717, 869197, 1051503, 987045, 1333241, 1190131, 1843049, 2991447, 3073340, 4382249, 4630456, 4635744, 5914142, 6877208

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 09 2020

See A332978 for the Pythagorean triple ordering and the links for images of the triangles.

Cf. A332978 (regions), A333135 (n-gons), A333137 (edges), A103605 (Pythagorean triple ordering), A092866, A332599, A007569.

a(8)-a(24) from Lars Blomberg, Jun 07 2020

A332610 Triangle read by rows: T(m,n) = number of triangular regions in a "frame" of size m X n with m >= n >= 1 (see Comments in A331457 for definition of frame).

Original entry on oeis.org

4, 14, 48, 32, 102, 128, 70, 192, 204, 288, 124, 326, 312, 396, 512, 226, 524, 516, 600, 716, 928, 360, 802, 784, 868, 984, 1196, 1472, 566, 1192, 1196, 1280, 1396, 1608, 1884, 2304, 820, 1634, 1704, 1788, 1904, 2116, 2392, 2812, 3328, 1218, 2296, 2500, 2584, 2700, 2912, 3188, 3608, 4124, 4928

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 12 2020

See A331457 for illustrations.

			Triangle begins:
[4],
[14, 48],
[32, 102, 128],
[70, 192, 204, 288],
[124, 326, 312, 396, 512],
[226, 524, 516, 600, 716, 928],
[360, 802, 784, 868, 984, 1196, 1472],
[566, 1192, 1196, 1280, 1396, 1608, 1884, 2304],
[820, 1634, 1704, 1788, 1904, 2116, 2392, 2812, 3328],
[1218, 2296, 2500, 2584, 2700, 2912, 3188, 3608, 4124, 4928],
[1696, 3074, 3456, 3540, 3656, 3868, 4144, 4564, 5080, 5884, 6848],
[2310, 4052, 4684, 4768, 4884, 5096, 5372, 5792, 6308, 7112, 8076, 9312],
...

Cf. A331457, A332599, A332600, A324042, A324043, A332606, A332607, A332595, A332596.

The first column is A324042, for which there is an explicit formula.
No formula is known for column 2, which is A332606.
For m>=n>=3 we have the (new) theorem that T(m,n) = 4*(m^2+n^2)+12*n+4*m-24 + 4*V(m,m,2)+4*V(n,n,2), where V(m,n,q) = Sum_{i = 1..m, j = 1..n, gcd(i,j)=q} (m+1-i)*(n+1-j).

A332611 Triangle read by rows: T(m,n) = number of quadrilateral regions in a "frame" of size m X n with m >= n >= 1 (see Comments in A331457 for definition of frame).

Original entry on oeis.org

0, 2, 8, 14, 36, 80, 34, 92, 144, 208, 90, 194, 280, 356, 504, 154, 336, 432, 520, 680, 856, 288, 554, 724, 824, 996, 1184, 1512, 462, 812, 1096, 1208, 1392, 1592, 1932, 2352, 742, 1314, 1680, 1804, 2000, 2212, 2564, 2996, 3640, 1038, 1756, 2296, 2432, 2640, 2864, 3228, 3672, 4328, 5016

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 12 2020

See A331457 for illustrations.

			Triangle begins:
[0],
[2, 8],
[14, 36, 80],
[34, 92, 144, 208],
[90, 194, 280, 356, 504],
[154, 336, 432, 520, 680, 856],
[288, 554, 724, 824, 996, 1184, 1512],
[462, 812, 1096, 1208, 1392, 1592, 1932, 2352],
[742, 1314, 1680, 1804, 2000, 2212, 2564, 2996, 3640],
[1038, 1756, 2296, 2432, 2640, 2864, 3228, 3672, 4328, 5016],
[1512, 2508, 3268, 3416, 3636, 3872, 4248, 4704, 5372, 6072, 7128],
[2074, 3252, 4416, 4576, 4808, 5056, 5444, 5912, 6592, 7304, 8372, 9616],
....

Cf. A331457, A332599, A332600, A324042, A324043, A332606, A332607, A332595, A332596.

The first column is A324043, for which there is an explicit formula.
No formula is known for column 2, which is A332607.
For m>=n>=3 we have the (new) theorem that T(m,n) = 4*(3*m*n-m-4*n) + 2*(V(m,m,1)-2*V(m,m,2)-m^2-4*m+6) + 2*(V(n,n,1)-2*V(n,n,2)-n^2-4*n+6) where V(m,n,q) = Sum_{i = 1..m, j = 1..n, gcd(i,j)=q} (m+1-i)*(n+1-j).

A338003 The number of vertices in a 4-pointed star 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

37, 653, 5517, 17153, 50349, 97037, 204329, 330613, 571021, 835713, 1298533, 1764125

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Oct 06 2020

See A338002 for a definition of the star and images for the number of regions.

Scott R. Shannon, Image of the star showing the vertex counts for n=1.
Scott R. Shannon, Image of the star showing the vertex counts for n=2.
Scott R. Shannon, Image of the star showing the vertex counts for n=3.
Scott R. Shannon, Image of the star showing the vertex counts for n=4.
Scott R. Shannon, Image of the star showing the vertex counts for n=5.

Cf. A338002 (number of regions), A092866, A332599, A007569, A333117, A333116.

a(8)-a(12) from Lars Blomberg, Apr 08 2021

cp's OEIS Frontend

A333113 The number of vertices inside a heptagon 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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A332608 Number of pentagons in the graph formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of a 3 X (n+1) rectangular grid of points (or equally, a 2 X n grid of squares).

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A333116 The number of vertices 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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A333117 The number of vertices inside a pentagram 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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A332609 Maximum number of edges in any cell in the graph formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of a 3 X (n+1) rectangular grid of points (or equally, a 2 X n grid of squares).

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A333136 The number of vertices formed on a triangle with leg lengths equal to the Pythagorean triples by the straight line segments mutually connecting all vertices and all points that divide the sides into unit length parts.

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A332610 Triangle read by rows: T(m,n) = number of triangular regions in a "frame" of size m X n with m >= n >= 1 (see Comments in A331457 for definition of frame).

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A332611 Triangle read by rows: T(m,n) = number of quadrilateral regions in a "frame" of size m X n with m >= n >= 1 (see Comments in A331457 for definition of frame).

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A338003 The number of vertices in a 4-pointed star 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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