A359061 Irregular table read by rows: T(n,k) is the number of k-gons formed, k>=2, among all circles that can be constructed on vertices of an n-sided regular polygon, using only a compass.

3, 0, 7, 0, 16, 29, 0, 30, 35, 1, 0, 90, 96, 0, 105, 126, 35, 1, 0, 272, 304, 48, 32, 0, 1, 0, 315, 324, 81, 0, 0, 0, 1, 0, 460, 940, 60, 40, 0, 0, 0, 1, 0, 671, 858, 264, 88, 11, 0, 0, 0, 1, 0, 960, 1656, 108, 48, 0, 1144, 1807, 559, 130, 13, 0, 0, 0, 0, 0, 1, 0, 1960, 3136, 448, 168, 0, 14, 0, 0, 0, 0, 0, 1

Offset: 2

Author

Scott R. Shannon, Dec 14 2022

Keywords

• nonn
• tabf

Comments

See A331702 and A359046 for further details and images.

Conjecture: the only value for n which leads to the creation of 2-gons is n = 2. Despite values for n mod 6 = 0 forming intersecting arcs at the center of the n-gon, these are cut by other circles and thus create 3-gons or 4-gons. This is in contrast to values of n mod 4 = 0 in A359009 which do lead to the creation of 2-gons at the center of the figure from similar arcs.

Examples

			The table begins:
3;
0, 7;
0, 16, 29;
0, 30, 35, 1;
0, 90, 96;
0, 105, 126, 35, 1;
0, 272, 304, 48, 32, 0, 1;
0, 315, 324, 81, 0, 0, 0, 1;
0, 460, 940, 60, 40, 0, 0, 0, 1;
0, 671, 858, 264, 88, 11, 0, 0, 0, 1;
0, 960, 1656, 108, 48;
0, 1144, 1807, 559, 130, 13, 0, 0, 0, 0, 0, 1;
0, 1960, 3136, 448, 168, 0, 14, 0, 0, 0, 0, 0, 1;
0, 2100, 3270, 945, 180, 15, 0, 0, 0, 0, 0, 0, 0, 1;
0, 3088, 5584, 896, 368, 16, 16, 0, 0, 0, 0, 0, 0, 0, 1;
0, 3400, 5814, 1513, 493, 85, 34, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 4536, 8712, 1224, 288, 54, 36;
0, 5586, 8797, 2774, 665, 76, 152, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 7940, 12480, 2440, 960, 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 7833, 14175, 3486, 1050, 147, 63, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 10428, 19448, 3850, 1408, 22, 44, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
		

Links

• Scott R. Shannon, Image for n = 17.
• Scott R. Shannon, Image for n = 23.
• Scott R. Shannon, Image for n = 24.

Crossrefs

Cf. A331702 (vertices), A359046 (regions), A359047 (edges), A359009, A358782, A007678.

Formula

Sum of row n = A359046(n).