login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A169808 Array T(n,k) read by antidiagonals: T(n,k) is the number of [n,k]-triangulations in the plane, n >= 0, k >= 0. 12
1, 1, 1, 1, 2, 1, 3, 4, 5, 4, 4, 11, 14, 18, 16, 12, 28, 53, 69, 88, 78, 27, 91, 178, 295, 396, 489, 457, 82, 291, 685, 1196, 1867, 2503, 3071, 2938, 228, 1004, 2548, 5051, 8385, 12560, 16905, 20667, 20118, 733, 3471, 9876, 21018, 38078, 60736, 89038, 119571, 146381, 144113 (list; table; graph; refs; listen; history; text; internal format)
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.
T(n,k) is the number of floor plan arrangements represented by 3-connected trivalent maps with n internal rooms and k+3 rooms adjacent to the outside.
"... may be evaluated from the results given by Brown."
The initial terms of this sequence can also be computed using the tool "plantri", in particular the command "./plantri -u -v -P -c2m2 [n]" will compute values for a diagonal. The '-c2' and '-m2' options indicate graphs must be biconnected and with minimum vertex degree 2. - Andrew Howroyd, Feb 22 2021
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
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy].
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)
Andrew Howroyd, PARI Program
FORMULA
T(n,k) = (A262586(n,k) + A169809(n,k)) / 2. - Andrew Howroyd, Feb 22 2021
EXAMPLE
Array begins:
============================================================
n\k | 0 1 2 3 4 5 6
----+-------------------------------------------------------
0 | 1 1 1 3 4 12 27 ...
1 | 1 2 4 11 28 91 291 ...
2 | 1 5 14 53 178 685 2548 ...
3 | 4 18 69 295 1196 5051 21018 ...
4 | 16 88 396 1867 8385 38078 169918 ...
5 | 78 489 2503 12560 60736 290595 1367374 ...
6 | 457 3071 16905 89038 451613 2251035 11025626 ...
7 | 2938 20667 119571 652198 3429943 17658448 89328186 ...
...
PROG
(PARI) \\ See link for script file.
A169808Array(6) \\ Andrew Howroyd, Feb 22 2021
CROSSREFS
Columns k=0..3 are A002713, A005500, A005501, A005502.
Rows n=0..2 are A000207, A005503, A005504.
Antidiagonal sums give A005027.
Cf. A146305 (rooted), A169809 (achiral), A262586 (oriented).
Sequence in context: A279436 A082470 A101204 * A328395 A283069 A304528
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, May 25 2010
EXTENSIONS
Edited by Andrew Howroyd, Feb 22 2021
a(29) corrected and terms a(36) and beyond from Andrew Howroyd, Feb 22 2021
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 13:41 EDT 2024. Contains 371254 sequences. (Running on oeis4.)