A005501 -id:A005501 - cp's OEIS frontend

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.

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

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.
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

			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 ...
  ...

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
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.
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

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).

\\ See link for script file.
A169808Array(6) \\ Andrew Howroyd, Feb 22 2021

T(n,k) = (A262586(n,k) + A169809(n,k)) / 2. - Andrew Howroyd, Feb 22 2021

Edited by Andrew Howroyd, Feb 22 2021

a(29) corrected and terms a(36) and beyond from Andrew Howroyd, Feb 22 2021

A378103 Triangle read by rows: T(n,k) is the number of n-node connected unsensed planar maps with an external face and k triangular internal faces, n >= 3, 1 <= k <= 2*n - 5.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 0, 2, 4, 4, 5, 4, 0, 0, 2, 6, 10, 14, 14, 18, 16, 0, 0, 0, 7, 18, 35, 49, 63, 69, 88, 78, 0, 0, 0, 5, 28, 74, 131, 204, 274, 345, 396, 489, 457, 0, 0, 0, 0, 26, 126, 304, 574, 893, 1290, 1708, 2137, 2503, 3071, 2938, 0, 0, 0, 0, 13, 159, 582, 1396, 2613, 4274, 6270, 8709, 11433, 14227, 16905, 20667, 20118

Offset: 3

Ya-Ping Lu, Nov 16 2024

The planar maps considered are without loops or isthmuses.
In other words, a(n) is the number of embeddings in the plane of connected bridgeless planar simple graphs with n vertices and k triangular internal faces.
The number of edges is n + k - 1.
The nonzero terms in row n range from k = floor(n/2) through 2*n-5 and, thus, the number of nonzero terms is 2n - floor(n/2) - 4 = A001651(n-2).

			Triangle begins:
n\k        1     2     3     4     5     6     7     8     9    10    11
----     ----  ----  ----  ----  ----  ----  ----  ----  ----  ----  ----
3          1
4          0     1     1
5          0     1     1     2     1
6          0     0     2     4     4     5     4
7          0     0     2     6    10    14    14    18    16
8          0     0     0     7    18    35    49    63    69    88    78

Andrew Howroyd, Table of n, a(n) for n = 3..2306 (rows 3..50)
Ya-Ping Lu, Illustration of initial terms

Row sums are A377785.
Cf. A001651, A002713, A003094, A169808, A378336 (sensed), A378340 (achiral).
The final 3 terms of each row are in A002713, A005500, A005501.

my(A=A378103rows(10)); for(i=1, #A, print(A[i])) \\ See PARI link in A378340 for program code. - Andrew Howroyd, Nov 25 2024

T(n, 2*n-5) = A002713(n-3).

T(n,k) = (A378336(n,k) + A378340(n,k))/2.

a(39) onwards from Andrew Howroyd, Nov 25 2024

A005502 Number of unrooted triangulations of a hexagon with n internal nodes.

Original entry on oeis.org

3, 11, 53, 295, 1867, 12560, 89038, 652198, 4903955, 37627699, 293607612, 2323604832, 18614121391, 150704813812, 1231596828200, 10148762396401, 84252059397251, 704122279126074, 5920239345451780, 50051285956517452, 425273487358680290, 3630084126997807369

Offset: 0

N. J. A. Sloane

nonn

These are also called [n,3]-triangulations.

Graphs can be enumerated and counted using the tool "plantri", in particular the command "./plantri -s -P6 -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=3 of the array in A169808.

Cf. A005507, A005495.

a(n) = (A005507(n) + A005495(n))/2 (based on Max Alekseyev's formula, cf. A005501 and A005500).

a(5)-a(10) from Manfred Scheucher, Mar 08 2018

Name clarified and terms a(11) and beyond from Andrew Howroyd, Feb 22 2021

A005506 Number of unrooted triangulations with reflection symmetry of a pentagon with n internal nodes.

Original entry on oeis.org

1, 3, 7, 19, 57, 176, 557, 1806, 5954, 19897, 67235, 229366, 788688, 2730810, 9512107, 33309444, 117190184, 414039578, 1468349782, 5225201321, 18651958885, 66769742002, 239643164237, 862168692562, 3108716586702, 11232127258416, 40660388117380, 147453014455094

Offset: 0

N. J. A. Sloane

nonn

These are also called [n,2]-triangulations.

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
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=2 of the array in A169809.

Cf. A002711, A005501.

a(n) = 2 * A005501(n) - A002711(n) (based on Max Alekseyev's formula, cf. A005501).

a(6)-a(11) from Altug Alkan and Manfred Scheucher, Mar 08 2018

Name clarified and terms a(12) and beyond from Andrew Howroyd, Feb 21 2021

A005507 Number of unrooted triangulations with reflection symmetry of a hexagon with n internal nodes.

Original entry on oeis.org

2, 6, 18, 52, 166, 524, 1722, 5664, 19072, 64408, 220676, 758864, 2634734, 9180872, 32208376, 113371636, 401067522, 1423073892, 5068961452, 18103192360, 64853607912, 232872927444, 838311889890, 3023961593292, 10931277735230, 39586258360246, 143617299291242

Offset: 0

N. J. A. Sloane

nonn

These are also called [n,3]-triangulations.

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
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=3 of the array in A169809.

Cf. A005495, A005502.

a(n) = 2 * A005502(n) - A005495(n) (based on Max Alekseyev's formula, cf. A005500 and A005501).

a(5)-a(10) from Altug Alkan and Manfred Scheucher, Mar 08 2018

Name clarified and terms a(11) and beyond from Andrew Howroyd, Feb 21 2021

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Formula

Extensions

A378103 Triangle read by rows: T(n,k) is the number of n-node connected unsensed planar maps with an external face and k triangular internal faces, n >= 3, 1 <= k <= 2*n - 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A005502 Number of unrooted triangulations of a hexagon with n internal nodes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions

A005506 Number of unrooted triangulations with reflection symmetry of a pentagon with n internal nodes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions

A005507 Number of unrooted triangulations with reflection symmetry of a hexagon with n internal nodes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions