A341800 Irregular table read by rows: Take a 2 by 1 ellipse with all diagonals drawn, as in A341688. Then T(n,k) = number of k-sided polygons in the figure containing 2n vertices, for k >= 3.

0, 4, 18, 6, 52, 28, 4, 120, 84, 22, 6, 244, 192, 72, 16, 434, 432, 124, 54, 8, 748, 748, 300, 52, 16, 4, 1234, 1232, 482, 164, 26, 2, 4, 1896, 1940, 776, 220, 36, 8, 2764, 2926, 1332, 330, 78, 10, 3892, 4460, 1716, 536, 88, 28, 0, 4, 5580, 5918, 2642, 784, 152, 44, 4

Offset

1,2

Comments

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

See A341688 for a description of the ellipse and images of the regions, and A341762 for images of the vertices.

Links

Table of n, a(n) for n=1..60.

Scott R. Shannon, Image of the n-gons for n = 6.

Wikipedia, Ellipse.

Formula

Row sums = A341688(n).

Example

A 2 by 1 ellipse consisting of 12 vertices, n = 6, contains 244 triangle, 192 quadrilaterals, 72 pentagons, 16 hexagons and no other n-gons, so the sixth row is [244, 192, 72, 16]. See the linked image.
The table begins:
0;
4;
18, 6;
52, 28, 4;
120, 84, 22, 6;
244, 192, 72, 16;
434, 432, 124, 54, 8;
748, 748, 300, 52, 16, 4;
1234, 1232, 482, 164, 26, 2, 4;
1896, 1940, 776, 220, 36, 8;
2764, 2926, 1332, 330, 78, 10;
3892, 4460, 1716, 536, 88, 28, 0, 4;
5580, 5918, 2642, 784, 152, 44, 4;
7508, 8204, 3540, 1108, 224, 12, 4, 4;
9902, 11202, 4636, 1472, 362, 44, 10, 4;
12984, 14508, 6208, 1920, 412, 80, 12;
16804, 18396, 8272, 2522, 522, 136, 20;
21212, 23352, 10580, 3144, 672, 112, 36;
26602, 28938, 13438, 4264, 766, 162, 12, 2;
32732, 36200, 16124, 5276, 952, 192, 12;
40026, 44216, 20038, 6564, 1302, 216, 16, 2;

Crossrefs

Cf. A341688 (regions), A341762 (vertices), A341764 (edges), A007678, A092867, A255011, A331929, A331931, A333075.

Sequence in context: A204936 A158320 A342153 * A236563 A264207 A264003

Adjacent sequences: A341797 A341798 A341799 * A341801 A341802 A341803

Keyword

nonn,tabf

Author

Scott R. Shannon and N. J. A. Sloane, Feb 20 2021

Status

approved