A290867
Irregular triangle read by rows: the number of points that are the intersections of k semicircles in the configuration A290447(n).
Original entry on oeis.org
0, 0, 0, 0, 1, 0, 5, 0, 15, 0, 35, 0, 70, 0, 123, 1, 0, 195, 5, 0, 285, 15, 0, 420, 25, 0, 586, 39, 2, 0, 818, 53, 4, 0, 1110, 73, 6, 0, 1451, 103, 10, 0, 1846, 142, 18, 0, 2361, 181, 26, 0, 2956, 234, 33, 2, 0, 3704, 287, 40, 4, 0, 4567, 348, 49, 8
Offset: 1
Triangle begins:
0;
0;
0;
0, 1;
0, 5;
0, 15;
0, 35;
0, 70;
0, 123, 1;
0, 195, 5;
0, 285, 15;
0, 420, 25;
0, 586, 39, 2;
- David Applegate, Table of n, a(n) for n = 1..800
- David Applegate, Triangular table T(n,k) for n = 1..100
- N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)
- 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.
A329713
The number of regions inside a heptagon 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
50, 868, 5594, 18396, 48462, 101794, 195714, 336504, 549704, 841890, 1249676, 1774612, 2468572, 3328234, 4414054, 5725034, 7336855, 9233098, 11513419, 14149296, 17254434, 20805554, 24928380, 29573348, 34902155, 40861422, 47613161
Offset: 1
A331932
Triangle read by rows: Take a hexagon with all diagonals drawn, as in A331931. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n+4.
Original entry on oeis.org
18, 6, 0, 264, 108, 36, 0, 1344, 654, 252, 12, 6, 4164, 2772, 1020, 228, 24, 0, 10038, 7758, 2424, 516, 72, 24, 0, 21108, 16188, 6060, 1128, 156, 0, 0, 0, 39690, 32022, 13368, 3654, 432, 48, 0, 0, 0, 68052, 56616, 22980, 6084, 888, 120, 12, 0, 0, 0
Offset: 1
A hexagon with no other points along its edges, n = 1, contains 18 triangles, 6 quadrilaterals and no other n-gons, so the first row is [18,6,0]. A hexagon with 1 point dividing its edges, n = 2, contains 264 triangles, 108 quadrilaterals, 36 pentagons and no other n-gons, so the second row is [264,108,36,0].
Triangle begins:
18,6,0
264,108,36,0
1344,654,252,12,6
4164,2772,1020,228,24,0
10038,7758,2424,516,72,24,0
21108,16188,6060,1128,156,0,0,0
39690,32022,13368,3654,432,48,0,0,0
68052,56616,22980,6084,888,120,12,0,0,0
The row sums are A331931.
A332421
The number of regions inside a nonagon 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
154, 2754, 16858, 55098, 142318, 298350, 568162, 975294, 1585666, 2426292, 3588508, 5093604, 7067422, 9523746, 12612214, 16351218, 20924029, 26326026, 32789107, 40289238, 49093282, 59181228, 70852528
Offset: 1
A332606
Number of triangles 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
14, 48, 102, 192, 326, 524, 802, 1192, 1634, 2296, 3074, 4052, 5246, 6740, 8398, 10440, 12770, 15512, 18782, 22384, 26386, 31204, 36482, 42232, 48826, 56508, 64318, 73356, 83366, 93996, 106010, 118788, 132634, 148600, 164814, 182648, 201998, 223172, 245634
Offset: 1
Cf.
A331452,
A331453,
A331454,
A331763,
A331765,
A331766,
A332599,
A332600,
A331457,
A332607,
A332608,
A332609.
A332607
Number of quadrilaterals 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
2, 8, 36, 92, 194, 336, 554, 812, 1314, 1756, 2508, 3252, 4348, 5464, 7054, 8760, 11050, 13324, 16162, 19256, 23188, 27120, 32098, 37396, 43456, 49516, 57608, 65440, 74670, 84388, 95674, 107656, 120990, 133996, 150144, 166424, 185090, 203960, 224926, 247120
Offset: 1
Cf.
A331452,
A331453,
A331454,
A331763,
A331765,
A331766,
A332599,
A332600,
A331457,
A332606,
A332608,
A332609.
A333026
The number of vertices formed on an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.
Original entry on oeis.org
3, 6, 16, 45, 111, 230, 448, 769, 1229, 1858, 2860, 4007, 5737, 7724, 10115, 13074, 17172, 21454, 27288, 33332, 40413, 48944, 59594, 70213, 82983, 97608, 113672, 131032, 152986, 174088, 201090, 228295, 258467, 292726, 328080, 365633, 412291, 460834, 512016
Offset: 1
A333139
The number of regions inside a decagon 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
220, 4220, 25220, 84280, 217800, 456640, 873090, 1501520, 2436020, 3736540, 5523970, 7830800, 10879460, 14665340, 19398660, 25173960, 32203320, 40502280, 50458120, 61995140, 75517160
Offset: 1
Original entry on oeis.org
16, 56, 176, 388, 822, 1452, 2516, 3952, 6060, 8736, 12492, 17040, 23102, 30280, 39234, 49688, 62730, 77556, 95642, 115992, 139874, 166560, 197992, 232600, 272574, 316460, 366390, 420792, 482748, 549516, 624962, 706436, 796766, 893844, 1001074, 1115428
Offset: 1
- Lars Blomberg, Table of n, a(n) for n = 1..200
- Lars Blomberg, Colored illustration of a(1)
- Lars Blomberg, Colored illustration of a(2)
- Lars Blomberg, Colored illustration of a(3)
- Lars Blomberg, Colored illustration of a(4)
- Lars Blomberg, Colored illustration of a(5)
- Lars Blomberg, Colored illustration of a(6)
- Lars Blomberg, Colored illustration of a(7)
- Lars Blomberg, Colored illustration of a(8)
- Lars Blomberg, Colored illustration of a(9)
- Hugo Pfoertner, Illustrations of Chamber Complexes up to 5 X 5.
Original entry on oeis.org
28, 92, 296, 652, 1408, 2470, 4312, 6774, 10428, 14992, 21492, 29328, 39876, 52184, 67616, 85588, 108192, 133674, 164992, 200158, 241560, 287428, 341768, 401472, 470764, 546230, 632404, 726170, 833420, 948550, 1079204, 1220054, 1376552, 1543742, 1729000
Offset: 1
Comments