A002712 -id:A002712

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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.

+10
10

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

(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.

"... 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).

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

A002713

Number of unrooted triangulations of the disk with n interior nodes and 3 nodes on the boundary.
(Formerly M3524 N1431)

+10
4

1, 1, 1, 4, 16, 78, 457, 2938, 20118, 144113, 1065328, 8068332, 62297808, 488755938, 3886672165, 31269417102, 254141551498, 2084129777764, 17228043363781, 143432427097935, 1201853492038096, 10129428318995227, 85826173629557200

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

These are also called [n,0]-triangulations.

REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..500

W. G. Brown, Enumeration of triangulations of the disk, Proc. London Math. Soc., 14 (1964), 746-768.

W. G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]

CombOS - Combinatorial Object Server, generate planar graphs

FORMULA

a(n) = (A002709(n) + A002712(n)) / 2.

CROSSREFS

Column k=0 of A169808.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Terms a(9) onward from Max Alekseyev, May 11 2010

Name clarified by Andrew Howroyd, Feb 24 2021

STATUS

approved

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