A341800 - OEIS

%I #10 Feb 20 2021 13:34:59

%S 0,4,18,6,52,28,4,120,84,22,6,244,192,72,16,434,432,124,54,8,748,748,

%T 300,52,16,4,1234,1232,482,164,26,2,4,1896,1940,776,220,36,8,2764,

%U 2926,1332,330,78,10,3892,4460,1716,536,88,28,0,4,5580,5918,2642,784,152,44,4

%N 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.

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

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

%H Scott R. Shannon, <a href="/A341800/a341800.png">Image of the n-gons for n = 6</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ellipse">Ellipse</a>.

%F Row sums = A341688(n).

%e 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.

%e The table begins:

%e 0;

%e 4;

%e 18, 6;

%e 52, 28, 4;

%e 120, 84, 22, 6;

%e 244, 192, 72, 16;

%e 434, 432, 124, 54, 8;

%e 748, 748, 300, 52, 16, 4;

%e 1234, 1232, 482, 164, 26, 2, 4;

%e 1896, 1940, 776, 220, 36, 8;

%e 2764, 2926, 1332, 330, 78, 10;

%e 3892, 4460, 1716, 536, 88, 28, 0, 4;

%e 5580, 5918, 2642, 784, 152, 44, 4;

%e 7508, 8204, 3540, 1108, 224, 12, 4, 4;

%e 9902, 11202, 4636, 1472, 362, 44, 10, 4;

%e 12984, 14508, 6208, 1920, 412, 80, 12;

%e 16804, 18396, 8272, 2522, 522, 136, 20;

%e 21212, 23352, 10580, 3144, 672, 112, 36;

%e 26602, 28938, 13438, 4264, 766, 162, 12, 2;

%e 32732, 36200, 16124, 5276, 952, 192, 12;

%e 40026, 44216, 20038, 6564, 1302, 216, 16, 2;

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

%K nonn,tabf

%O 1,2

%A _Scott R. Shannon_ and _N. J. A. Sloane_, Feb 20 2021