A331907 -id:A331907 - cp's OEIS frontend

A331906 The number of regions inside a pentagram formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

40, 1100, 7330, 25540, 65930, 136200, 263010, 458410, 740550, 1142740, 1681640, 2400970, 3338850, 4495510, 5962220, 7736150, 9924580, 12442880, 15527670, 19132140, 23301600, 28070620, 33585800, 39919140, 47157510, 55209750, 64185300, 74311940, 85731780, 98167130

Offset: 1

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

nonn

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

Scott R. Shannon, Pentagram regions for n = 1.
Scott R. Shannon, Pentagram regions for n = 2.
Scott R. Shannon, Pentagram regions for n = 3.
Scott R. Shannon, Pentagram regions for n = 4.
Scott R. Shannon, Pentagram regions for n = 5.
Scott R. Shannon, Pentagram regions for n = 6.
Scott R. Shannon, Pentagram regions with random distance-based coloring for n = 1.
Scott R. Shannon, Pentagram regions with random distance-based coloring for n = 2.
Scott R. Shannon, Pentagram regions with random distance-based coloring for n = 3.
Scott R. Shannon, Pentagram regions with random distance-based coloring for n = 4.
Scott R. Shannon, Pentagram regions with random distance-based coloring for n = 5.
Eric Weisstein's World of Mathematics, Pentagram.

Cf. A331907 (n-gons), A333117 (vertices), A333118 (edges), A007678, A092867, A331452.

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

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

A333118 The number of edges 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

65, 1965, 14100, 49760, 130235, 268900, 522770, 911425, 1474680, 2276820, 3354695, 4785575, 6665240, 8973795, 11903415, 15446945, 19825715, 24850460, 31025390, 38221130, 46557865, 56092005, 67123385, 79765335, 94249750, 110346520, 128289075, 148525930, 171374335, 196206590

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), A333117 (vertices), A274586, A332600, A331765.

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

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A331906 The number of regions 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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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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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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A333118 The number of edges 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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