A210696 -id:A210696 - cp's OEIS frontend

A005497 Erroneous version of A210696.

Original entry on oeis.org

1, 2, 5, 16, 48, 164, 599, 1952

Offset: 0

N. J. A. Sloane

dead

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

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

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.

Original entry on oeis.org

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.

A002711 Triangulations of the disk G_{n,2}.

Original entry on oeis.org

1, 5, 21, 119, 735, 4830, 33253, 237336, 1743588, 13114465, 100591491, 784428522, 6204258970, 49675571820, 402013608525, 3284214703056, 27055525967324, 224557688302164, 1876374953765620, 15774315150669075, 133345566025874055, 1132905339504465590, 9669675838370401035

Offset: 0

N. J. A. Sloane

nonn

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).

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

Column k=2 of A262586.

Cf. A210696.

Maple
```
# see A210696
```

Extended by Max Alekseyev, Mar 30 2009

a(21) onwards from Andrew Howroyd, Nov 23 2024

A005498 Triangulations of the disk G_{2,n}.

Original entry on oeis.org

1, 6, 21, 88, 330, 1302, 5005, 19504, 75582, 294140, 1144066, 4458192, 17383860, 67866918, 265182525, 1037169760, 4059928950, 15905412468, 62359143990, 244662838160, 960566918220, 3773656396796, 14833897694226, 58343359313568, 229591913401900, 903936171565752, 3560597348629860

Offset: 0

N. J. A. Sloane

nonn

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..500
Jean-François Alcover, Mathematica program
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]

Row n=2 of A262586.

A005498 := proc(n)
        BrownG(2,n) ;
end proc:  # take BrownG from A210696
seq(A005498(n),n=0..25) ; # R. J. Mathar, Mar 30 2012

Mathematica
```
(* See links *)
```

More terms from R. J. Mathar, Mar 30 2012

a(25) onwards from Andrew Howroyd, Nov 23 2024

A005495 Number of triangulations of the disk G_{n,3}.

Original entry on oeis.org

4, 16, 88, 538, 3568, 24596, 176354, 1298732, 9788838, 75190990, 586994548, 4646450800, 37225608048, 301400446752, 2463161448024, 20297411421166, 168503717726980, 1408243135178256, 11840473621942108, 100102553809842544

Offset: 0

N. J. A. Sloane

nonn

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

William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.

A005495 := proc(n) # see A210696
        BrownG(n,3) ;
end proc:
seq(A005495(n),n=0..25) ; # R. J. Mathar, Mar 30 2012

(* See A210696 for BrownG[n,m]  *)
a[n_] := BrownG[n, 3];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 25 2023 *)

A005499 Triangulations of the disk G_{3,n}.

Original entry on oeis.org

5, 26, 119, 538, 2310, 9882, 41715, 175088, 730626, 3037510, 12584726, 52003792, 214401024, 882233898, 3624161175, 14865947668, 60898934250, 249184153548, 1018532686314, 4159265561360, 16970015555220, 69183689403686, 281844056190294, 1147419353238816, 4668368905854840, 18982659409726792

Offset: 0

N. J. A. Sloane

nonn

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..500
Jean-François Alcover, Mathematica program
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]

Row n=3 of A262586.

A005499 := proc(n)
        BrownG(3,n) ;
end proc:  # take BrownG from A210696
seq(A005499(n),n=0..25) ; # R. J. Mathar, Mar 30 2012

Mathematica
```
(* See links *)
```

a(22) onwards from Andrew Howroyd, Nov 23 2024

cp's OEIS Frontend

A005497 Erroneous version of A210696.

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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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A002711 Triangulations of the disk G_{n,2}.

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Maple

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A005498 Triangulations of the disk G_{2,n}.

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Maple

Mathematica

Extensions

A005495 Number of triangulations of the disk G_{n,3}.

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Programs

Maple

Mathematica

A005499 Triangulations of the disk G_{3,n}.

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Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Extensions