A169809 Array T(n,k) read by antidiagonals: T(n,k) is the number of [n,k]-triangulations in the plane that have reflection symmetry, n >= 0, k >= 0.

1, 1, 1, 1, 2, 1, 2, 3, 4, 3, 2, 6, 7, 10, 8, 5, 8, 18, 19, 29, 23, 5, 18, 26, 52, 57, 86, 68, 14, 23, 68, 82, 166, 176, 266, 215, 14, 56, 91, 220, 270, 524, 557, 844, 680, 42, 70, 248, 321, 769, 890, 1722, 1806, 2742, 2226, 42, 180, 318, 872, 1151, 2568, 2986, 5664, 5954, 9032, 7327

Offset

0,5

Comments

"A closed bounded region in the plane divided into triangular regions with k+3 vertices on the boundary and n internal vertices is said to be a triangular map of type [n,k]." It is a [n,k]-triangulation if there are no multiple edges.

"... may be evaluated from the results given by Brown."

References

C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325

William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.

C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979. (Annotated scanned copy)

Example

Array begins:
====================================================
n\k |   0   1    2    3     4     5     6      7
----+-----------------------------------------------
  0 |   1   1    1    2     2     5     5     14 ...
  1 |   1   2    3    6     8    18    23     56 ...
  2 |   1   4    7   18    26    68    91    248 ...
  3 |   3  10   19   52    82   220   321    872 ...
  4 |   8  29   57  166   270   769  1151   3296 ...
  5 |  23  86  176  524   890  2568  4020  11558 ...
  6 |  68 266  557 1722  2986  8902 14197  42026 ...
  7 | 215 844 1806 5664 10076 30362 49762 148208 ...
  ...

Prog

(PARI) \\ See link in A169808 for script.
A169809Array(7) \\ Andrew Howroyd, Feb 22 2021

Crossrefs

Columns k=0..3 are A002712, A005505, A005506, A005507.

Rows n=0..2 are A208355, A005508, A005509.

Antidiagonal sums give A005028.

Cf. A146305 (rooted), A169808 (unrooted), A262586 (oriented).

Sequence in context: A196686 A213088 A357564 * A214962 A350809 A366525

Adjacent sequences: A169806 A169807 A169808 * A169810 A169811 A169812

Keyword

nonn,tabl

Author

N. J. A. Sloane, May 25 2010

Extensions

Edited and terms a(36) and beyond from Andrew Howroyd, Feb 22 2021

Status

approved