A002711 -id:A002711 - cp's OEIS frontend

A262586 Square array T(n,m) (n>=0, m>=0) read by antidiagonals downwards giving number of rooted triangulations of type [n,m] up to orientation-preserving isomorphisms.

1, 1, 1, 1, 2, 1, 4, 5, 6, 5, 6, 16, 21, 26, 24, 19, 48, 88, 119, 147, 133, 49, 164, 330, 538, 735, 892, 846, 150, 559, 1302, 2310, 3568, 4830, 5876, 5661, 442, 1952, 5005, 9882, 16500, 24596, 33253, 40490, 39556, 1424, 6872, 19504, 41715, 75387, 120582, 176354, 237336, 290020, 286000, 4522

Offset: 0

N. J. A. Sloane, Oct 20 2015

			Array begins:
 ==============================================================
 n\k |    0     1      2       3       4        5         6 ...
 ----+---------------------------------------------------------
   0 |    1     1      1       4       6       19        49 ...
   1 |    1     2      5      16      48      164       559 ...
   2 |    1     6     21      88     330     1302      5005 ...
   3 |    5    26    119     538    2310     9882     41715 ...
   4 |   24   147    735    3568   16500    75387    338685 ...
   5 |  133   892   4830   24596  120582   578622   2730728 ...
   6 |  846  5876  33253  176354  900240  4493168  22037055 ...
   7 | 5661 40490 237336 1298732 6849810 35286534 178606610 ...
   ...
The first few antidiagonals are:
  1,
  1,1,
  1,2,1,
  4,5,6,5,
  6,16,21,26,24,
  19,48,88,119,147,133,
  49,164,330,538,735,892,846,
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).
Jean-François Alcover, Mathematica code
W. G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]. See Table 1 (with a typo at G(n=1,m=6)).
L. March and C. F. Earl, On Counting Architectural Plans, Environment and Planning B, 4 (1977), 57-80. See Table 2.

Columns 0..2 are A002709, A002710, A002711.
Rows 0..3 are A001683, A210696, A005498, A005499.
Antidiagonal sums are A341855.
Cf. A169808 (unoriented), A169809 (achiral).

A262586 := proc(n,m)
    BrownG(n,m) ; # procedure in A210696
end proc:
for d from 0 to 12 do
    for n from 0 to d do
        printf("%d,",A262586(n,d-n)) ;
    end do:
end do: # R. J. Mathar, Oct 21 2015

Mathematica
```
(* See LINKS section. *)
```

\\ See Links in A169808 for PARI program file.
{ for(n=0, 7, for(k=0, 7, print1(OrientedTriangs(n,k), ", ")); print) } \\ Andrew Howroyd, Nov 23 2024

Brown (Eq. 6.3) gives a formula.

A005501 Number of unrooted triangulations of a pentagon with n internal nodes.

Original entry on oeis.org

1, 4, 14, 69, 396, 2503, 16905, 119571, 874771, 6567181, 50329363, 392328944, 3102523829, 24839151315, 201011560316, 1642124006250, 13527821578754, 112279051170871, 938188211057701, 7887160187935198, 66672792338916470, 566452703137103796, 4834838039006782636

Offset: 0

N. J. A. Sloane

nonn

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

Graphs can be enumerated and counted using the tool "plantri", in particular the command "./plantri -s -P5 -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=2 of the array in A169808.

Cf. A002711, A005506.

a(n) = (A005506(n) + A002711(n))/2. - Max Alekseyev, Oct 29 2012

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

Name clarified and terms a(12) and beyond from Andrew Howroyd, Feb 22 2021

A378336 Triangle read by rows: T(n,k) is the number of n node connected sensed 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, 5, 5, 6, 5, 0, 0, 2, 8, 13, 20, 21, 26, 24, 0, 0, 0, 10, 28, 55, 79, 104, 119, 147, 133, 0, 0, 0, 7, 45, 126, 230, 360, 491, 625, 735, 892, 846, 0, 0, 0, 0, 44, 227, 561, 1066, 1682, 2430, 3241, 4074, 4830, 5876, 5661

Offset: 3

Andrew Howroyd, Nov 23 2024

See A378103 for illustration of initial terms. This sequence does not consider planar maps to be equivalent to their reflections.
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 up to orientation preserving isomorphisms.
The number of edges is n + k - 1.

			Triangle begins:
n\k | 1  2  3   4   5    6    7    8    9   10   11   12   13
----+----------------------------------------------------------
  3 | 1;
  4 | 0, 1, 1;
  5 | 0, 1, 1,  2,  1;
  6 | 0, 0, 2,  5,  5,   6,   5;
  7 | 0, 0, 2,  8, 13,  20,  21,  26,  24;
  8 | 0, 0, 0, 10, 28,  55,  79, 104, 119, 147, 133;
  9 | 0, 0, 0,  7, 45, 126, 230, 360, 491, 625, 735, 892, 846;
  ...

Andrew Howroyd, Table of n, a(n) for n = 3..2306 (rows 3..50)

Row sums are A378335.
Column sums are A378337.
Antidiagonal sums are A378338.
The final 3 terms of each row are in A002709, A002710, A002711.
Cf. A262586 (2-connected), A341923 (3-connected), A378103, (unsensed), A378340 (achiral).

my(A=A378336rows(10)); for(i=1, #A, print(A[i])) \\ See PARI link in A378340 for program code.

T(n,k) = 0 for n > 2*k + 1.
T(n,2*n-5) = A002709(n-3).
T(n,2*n-6) = A002710(n-4) for n >= 4.
T(n,2*n-7) = A002711(n-5) for n >= 5.

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

cp's OEIS Frontend

A262586 Square array T(n,m) (n>=0, m>=0) read by antidiagonals downwards giving number of rooted triangulations of type [n,m] up to orientation-preserving isomorphisms.

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A005501 Number of unrooted triangulations of a pentagon with n internal nodes.

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A378336 Triangle read by rows: T(n,k) is the number of n node connected sensed planar maps with an external face and k triangular internal faces, n >= 3, 1 <= k <= 2*n - 5.

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PARI

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

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