A342153 Irregular table read by rows: Take a vesica piscis with all diagonals drawn, as in A341877. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

0, 4, 18, 6, 52, 28, 4, 120, 78, 34, 4, 252, 188, 56, 12, 470, 348, 184, 40, 4, 2, 808, 648, 300, 56, 8, 1282, 1118, 548, 138, 20, 4, 2036, 1772, 644, 156, 28, 8, 2878, 2804, 1252, 388, 96, 10, 4172, 4024, 1728, 468, 100, 28, 5752, 5682, 2600, 866, 162, 46, 7912, 7676, 3420, 1024, 196, 44, 16

Offset

1,2

Comments

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

See A341877 for images of the regions and A341878 for images of the vertices.

Links

Table of n, a(n) for n=1..63.

Scott R. Shannon, Image of the k-gons for n=14.

Wikipedia, Vesica piscis.

Example

A vesica piscis with 1 point dividing its edges, n = 2, contains 4 triangles and no other n-gons, so the second row is [4]. A vesica piscis with 3 points dividing its edges, n = 4, contains 52 triangles, 28 quadrilaterals, 4 pentagons and no other n-gons, so the fourth row is [52, 28, 4].
The table begins:
0;
4;
18,6;
52,28,4;
120,78,34,4;
252,188,56,12;
470,348,184,40,4,2;
808,648,300,56,8;
1282,1118,548,138,20,4;
2036,1772,644,156,28,8;
2878,2804,1252,388,96,10;
4172,4024,1728,468,100,28;
5752,5682,2600,866,162,46;
7912,7676,3420,1024,196,44,16;
10388,10354,4868,1548,352,60,6;
13496,13808,6016,1836,388,80,4,0,4;
17310,17590,8376,2672,564,122,16,2;
22012,22364,10160,3152,712,124,20,4;
27440,27956,13162,4432,964,172,24,2,4;
33784,34736,15588,4640,1096,120,28;

Crossrefs

Cf. A341877 (regions), A342152 (edges), A341878 (vertices), A331451, A331911, A340614, A340688.

Sequence in context: A205014 A204936 A158320 * A341800 A236563 A264207

Adjacent sequences: A342150 A342151 A342152 * A342154 A342155 A342156

Keyword

nonn,tabf

Author

Scott R. Shannon and N. J. A. Sloane, Mar 02 2021

Status

approved