A005505 -id:A005505 - cp's OEIS frontend

A005500 Number of unrooted triangulations of a quadrilateral with n internal nodes.

1, 2, 5, 18, 88, 489, 3071, 20667, 146381, 1072760, 8071728, 61990477, 484182622, 3835654678, 30757242535, 249255692801, 2038827903834, 16815060576958, 139706974995635, 1168468902294726, 9831504782276593, 83174244225508659, 707159273362126228, 6039827641569969225

Offset: 0

N. J. A. Sloane

nonn

These are also called [n,1]-triangulations.

Graphs can be enumerated and counted using the tool "plantri", in particular the command "./plantri -s -P4 -c2m2 [n]". - Manfred Scheucher, Mar 08 2018

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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..200
G. Brinkmann and B. McKay, Plantri (program for generation of certain types of planar graph)
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)
C. F. Earl & N. J. A. Sloane, Correspondence, 1980-1981

Column k=1 of A169808.

Cf. A002710, A005505.

a(n) = (A005505(n) + A002710(n))/2. - Max Alekseyev, Oct 29 2012

Edited by Max Alekseyev, Oct 29 2012
a(7)-a(12) from Manfred Scheucher, Mar 08 2018
Name clarified and terms a(13) and beyond from Andrew Howroyd, Feb 22 2021

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.

Original entry on oeis.org

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

N. J. A. Sloane, May 25 2010

"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."

			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 ...
  ...

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.

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)

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).

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

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

cp's OEIS Frontend

A005500 Number of unrooted triangulations of a quadrilateral with n internal nodes.

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