A367662 -id:A367662 - cp's OEIS frontend

A367663 Number of vertices formed in a hexagon by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the hexagon.

6, 13, 55, 223, 673, 1753, 3787, 6937, 11851, 19297, 30211, 42877, 63247, 86323, 114049, 155701, 204169, 251293, 329989, 400081, 487537, 607189, 742747, 858241, 1041835, 1231789, 1424647, 1667659, 1968427, 2165947, 2600779, 2939863, 3286159, 3784153, 4217395, 4687771, 5688804, 6316332

Offset: 1

Scott R. Shannon, Nov 26 2023

Keyword "look" is because of the linked images. - N. J. A. Sloane, Dec 01 2023

Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 9.

Cf. A367662 (regions), A367664 (edges), A367665 (k-gons), A355799.

a(n) = A367664(n) - A367662(n) + 1 by Euler's formula.

A367665 Irregular table read by rows: T(n,k) is the number of k-sided polygons, for n>=1 and k>=3, in a hexagon when straight line segments connect the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the hexagon.

Original entry on oeis.org

0, 0, 0, 1, 0, 6, 30, 12, 12, 126, 102, 6, 18, 384, 294, 78, 18, 0, 6, 888, 726, 300, 42, 12, 1824, 1500, 594, 216, 12, 3186, 3048, 1206, 186, 36, 5592, 4914, 1866, 588, 48, 6, 8736, 8298, 3108, 816, 108, 12, 13164, 12786, 5046, 1368, 180, 18, 19578, 17940, 7242, 1854, 144, 60, 26700, 27426, 10842, 2652, 324, 18

Offset: 1

Scott R. Shannon, Nov 26 2023

			The table begins:
0, 0, 0, 1;
0, 6;
30, 12, 12;
126, 102, 6, 18;
384, 294, 78, 18, 0, 6;
888, 726, 300, 42, 12;
1824, 1500, 594, 216, 12;
3186, 3048, 1206, 186, 36;
5592, 4914, 1866, 588, 48, 6;
8736, 8298, 3108, 816, 108, 12;
13164, 12786, 5046, 1368, 180, 18;
19578, 17940, 7242, 1854, 144, 60;
26700, 27426, 10842, 2652, 324, 18;
36972, 37134, 15000, 3462, 408, 72;
50316, 48552, 19398, 4830, 480, 96, 0, 12;
64398, 68388, 27018, 6006, 852, 132;
83052, 89268, 35178, 8526, 1140, 204, 12;
106956, 109938, 42600, 10218, 1116, 204, 24;
131664, 145332, 56622, 14160, 1908, 234, 12;
166338, 173514, 69426, 16716, 2004, 252;
202932, 215028, 82374, 20202, 2400, 324, 12;
.
.

Scott R. Shannon, Image for T(9,k).

Cf. A367662 (regions), A367663 (vertices), A367664 (edges), A355801.

Sum of row(n) = A367662(n).

A367664 Number of edges formed in a hexagon by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the hexagon.

Original entry on oeis.org

6, 18, 108, 474, 1452, 3720, 7932, 14598, 24864, 40374, 62772, 89694, 131208, 179370, 237732, 322494, 421548, 522348, 679920, 828330, 1010808, 1252932, 1527540, 1776846, 2146800, 2536038, 2939664, 3436632, 4040088, 4483548, 5335020, 6041532, 6773148, 7771926, 8678256, 9666300

Offset: 1

Scott R. Shannon, Nov 26 2023

nonn

See A367662 and A367663 for images of the hexagon.

Cf. A367662 (regions), A367663 (vertices), A367665 (k-gons), A355800.

a(n) = A367662(n) + A367663(n) - 1 by Euler's formula.

cp's OEIS Frontend

A367663 Number of vertices formed in a hexagon by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the hexagon.

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A367665 Irregular table read by rows: T(n,k) is the number of k-sided polygons, for n>=1 and k>=3, in a hexagon when straight line segments connect the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the hexagon.

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A367664 Number of edges formed in a hexagon by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the hexagon.

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