A342053 Array read by antidiagonals: T(n,k) is the number of unrooted 3-connected triangulations of a disk with n interior nodes and k nodes on the boundary, n >= 1, k >= 3.

1, 1, 1, 1, 2, 4, 1, 2, 8, 16, 1, 3, 12, 38, 78, 1, 3, 20, 73, 219, 457, 1, 4, 27, 140, 503, 1404, 2938, 1, 4, 39, 235, 1089, 3661, 9714, 20118, 1, 5, 51, 392, 2149, 8796, 27715, 70454, 144113, 1, 5, 68, 610, 4050, 19419, 72204, 214664, 527235, 1065328

Offset: 1

Author

Andrew Howroyd, Feb 26 2021

Keywords

• nonn
• tabl

Comments

For k >= 4, T(n,k) is the number of polyhedra with n+k vertices whose faces are all triangular, except one which is k-gonal.

The initial terms of this sequence can also be computed using the tool "plantri", in particular the command "./plantri -u -v -P [n]" will compute values for a diagonal.

Examples

			Array begins:
===================================================
n\k |     3     4      5      6       7       8
----+----------------------------------------------
  1 |     1     1      1      1       1       1 ...
  2 |     1     2      2      3       3       4 ...
  3 |     4     8     12     20      27      39 ...
  4 |    16    38     73    140     235     392 ...
  5 |    78   219    503   1089    2149    4050 ...
  6 |   457  1404   3661   8796   19419   40485 ....
  7 |  2938  9714  27715  72204  173779  393123 ...
  8 | 20118 70454 214664 596906 1538221 3723976 ...
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275
• G. Brinkmann and B. McKay, Plantri (program for generation of certain types of planar graph)
• William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
• Andrew Howroyd, PARI Program

Crossrefs

Columns k=3..6 are A002713, A058786(n+4), A342054, A342055.

Antidiagonal sums are A342056.

Cf. A169808 (2-connected), A341856 (rooted), A341923 (oriented).

Programs

PARI

A342053Array(8,6) \\ See links for program.