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.
Original entry on oeis.org
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, 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.
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
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.
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
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
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