Günter Rote - cp's OEIS frontend

A366409 Number of smooth convex lattice polygons with area n/2.

1, 1, 1, 3, 2, 4, 4, 6, 5, 7, 7, 9, 7, 12, 12, 15, 9, 15, 16, 18, 13, 23, 21, 24, 19, 26, 25, 30, 22, 39, 34, 34, 27, 46, 42, 41, 35, 60, 53, 56, 41, 63, 61, 62, 61, 91, 66, 72, 78, 111, 87, 86, 83, 135, 123, 111, 97, 142, 135, 156, 146, 176, 148, 186, 194, 206, 169, 200, 242, 313

Offset: 1

Günter Rote, Oct 09 2023

nonn

A lattice polygon is a polygon whose vertices have integer coordinates. (They belong to the integer lattice or grid Z x Z).

A convex lattice polygon is smooth if, for every vertex V, the adjacent lattice points on the two incident edges (which are not necessarily vertices) form together with V a triangle of area 1/2.

			Here is a smooth lattice polygon with k=6 vertices (V), 2 lattice points on edges (B), 2 interior lattice points (I), and area 5, shown as part of the grid: (The edges of the polygon are not drawn.)
     V--V--+--+--+
     |  |  |  |  |
     V--I--B--+--+
     |  |  |  |  |
     +--V--I--B--+
     |  |  |  |  |
     +--+--+--V--V
See Bogart et al., Theorem 32, and Appendix, p. 325, for a list of all 41 (convex) smooth lattice polygons with at most 12 lattice points, with figures.
The dataset of Balletti gives the complete set of 1530 polygons up to area 25. Beware that the vertices are not always listed in sorted (clockwise or counterclockwise) order around the polygon boundary.

Günter Rote, Table of n, a(n) for n = 1..300 (first 50 terms from Balletti (2021), Table 2 on p. 1114).
Gabriele Balletti, Enumeration of lattice polytopes by their volume, Discrete Comput. Geom., 65 (2021), 1087-1122.
Gabriele Balletti, Dataset of "small" lattice polytopes (2018).
T. Bogart, C. Haase, M. Hering, B. Lorenz, B. Nill, A. Paffenholz, G. Rote, F. Santos, and H. Schenck, Finitely many smooth d-polytopes with n lattice points, Israel Journal of Mathematics 207 (2015), 301-329; and arXiv version, arXiv:1010.3887 [math.AG], 2010-2013.
Günter Rote, Python program to count convex lattice polygons up to a given area (2023).
Günter Rote, Number of smooth lattice polygons of area at most 150, classified by the number k of vertices, the number B of lattice points on edges, and the number I of interior lattice points.

Cf. A187015 for lattice polygons without the smoothness restriction. Cf. A127709.

Python
```
# See the links section.
```

A365641 The minimum number of ways to label each triangle of a triangulation of an n-gon with one of its vertices so that different triangles get different labels (minimum taken over all triangulations).

Original entry on oeis.org

1, 3, 7, 14, 25, 41, 63, 92, 128, 173, 228, 293, 369, 458, 561, 676, 807, 955, 1119, 1300

Offset: 2

Günter Rote, Sep 14 2023

Originally proposed by Johan Wästlund, Aug 28 2007, as an equivalent formulation of A089187.

The "fan triangulations", where one vertex is connected to all other vertices, is optimal up to a(9). Starting from a(10)=128, other triangulations are better.

			a(4)=7. Suppose a 4-gon ABCD is triangulated with triangles ABC and ACD. If ABC is labeled B, then ACD can be given 3 possible labels, while if ABC is labeled A or C, only 2 labels are available for ACD and 3+2+2=7. - Johan Wästlund, Aug 28 2007

"""For a chosen "base edge" of a triangulated (n+2)-gon, (ul, ur, u2)
denotes the numbers of labelings where the left, the right, or
both vertices of the base edge have been used as labels.
The number of labelings where none of the basepoints is used is always 1.
tri[n] will contain the possible triplets (ul,ur,u2) for the
triangulations of an (n+2)-gon."""
tri = [{(0,0,0)}] # start with single edge (2-gon); no labels
def combine(u,v):
    (ul,ur,u2),(vl,vr,v2) = u,v # formula obtained by combining the cases
    return (1+vl+ur+ul, 1+vl+ur+vr, vr+ul+(ul+ur)*(vl+vr)+u2+v2 )
for n in range(1,18): # dynamic programming, requires large memory
    tri.append({combine(u,v) for k in range(n)
                for u in tri[k] for v in tri[n-k-1]})
print(", ".join(str(1+min(sum(t) for t in tr)) for tr in tri))

A365620 Number of integer grid points on the circle around (0,0) with radius A088959(n).

Original entry on oeis.org

4, 12, 20, 36, 60, 108, 180, 252, 324, 540, 756, 972, 1620, 2268, 2916, 4860, 6804, 8748, 14580, 20412, 26244, 43740, 61236, 72900, 78732, 102060, 131220, 183708, 218700, 236196, 306180, 393660, 551124, 656100, 708588, 918540

Offset: 1

Günter Rote, Sep 12 2023

nonn

Sequence of records of A046109 (first term 1 from A046109 is omitted).

See A071385 for radii that are not necessarily integers.

a(n) = 8*A088111(n) + 4.

A354046 Number of 4-dimensional chiral point groups of order n.

Original entry on oeis.org

1, 2, 3, 6, 4, 10, 5, 17, 8, 14, 7, 27, 8, 18, 14, 37, 10, 31, 11, 36, 18, 26, 13, 71, 17, 30, 22, 46, 16, 56, 17, 73, 26, 38, 26, 78, 20, 42, 30, 86, 22, 72, 23, 66, 42, 50, 25, 147, 30, 67, 38, 76, 28, 88, 38, 108, 42, 62, 31, 142, 32, 66, 55, 134, 44

Offset: 1

Laith Rastanawi and Günter Rote, May 16 2022

nonn

In other words, the number of (finite) subgroups of order n of the special orthogonal group SO(4).

John H. Conway and Derek A. Smith, On Quaternions and Octonions, CRC Press, 2003.
Patrick Du Val, Homographies, Quaternions and Rotations. Clarendon Press, 1964.

Laith Rastanawi and Günter Rote, Table of n, a(n) for n = 1..10000
Edouard Goursat, Sur les substitutions orthogonales et les divisions régulières de l'espace, Annales scientifiques de l'E.N.S. 3e série, 6:9-102, 1889.
A. C. Hurley, Finite rotation groups and crystal classes in four dimensions, Mathematical Proceedings of the Cambridge Philosophical Society, 47(4):650-661, 1951.
Laith Rastanawi and Günter Rote, Towards a Geometric Understanding of the 4-Dimensional Point Groups, arXiv preprint arXiv:2205.04965 [math.MG], 2022.
Laith Rastanawi and Günter Rote, Sage code used to generate the sequence.
W. Threlfall and H. Seifert, Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes, Math. Annalen, 104:1-70, 1931.
W. Threlfall and H. Seifert, Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes (Schluß), Math. Annalen, 107:543-586, 1933.

Cf. A353341.

A353341 Number of 4-dimensional point groups of order n.

Original entry on oeis.org

1, 4, 3, 11, 4, 14, 5, 29, 8, 18, 7, 41, 8, 22, 14, 56, 10, 37, 11, 51, 18, 30, 13, 100, 17, 34, 22, 60, 16, 64, 17, 98, 26, 42, 26, 101, 20, 46, 30, 112, 22, 80, 23, 80, 42, 54, 25, 189, 30, 73, 38, 91, 28, 96, 38, 133, 42, 66, 31, 170, 32, 70, 55, 167, 44

Offset: 1

Laith Rastanawi and Günter Rote, May 16 2022

nonn

In other words, the number of (finite) subgroups of order n of the orthogonal group O(4).

John H. Conway and Derek A. Smith, On Quaternions and Octonions, CRC Press, 2003.
Patrick Du Val, Homographies, Quaternions and Rotations. Clarendon Press, 1964.

Laith Rastanawi and Günter Rote, Table of n, a(n) for n = 1..10000
Edouard Goursat, Sur les substitutions orthogonales et les divisions régulières de l'espace, Annales scientifiques de l'E.N.S. 3e série, 6:9-102, 1889.
A. C. Hurley, Finite rotation groups and crystal classes in four dimensions, Mathematical Proceedings of the Cambridge Philosophical Society, 47(4):650-661, 1951.
Laith Rastanawi and Günter Rote, Towards a Geometric Understanding of the 4-Dimensional Point Groups, arXiv preprint arXiv:2205.04965 [math.MG], 2022.
Laith Rastanawi and Günter Rote, Sage code used to generate the sequence.
W. Threlfall and H. Seifert, Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes, Math. Annalen, 104:1-70, 1931.
W. Threlfall and H. Seifert, Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes (Schluß), Math. Annalen, 107:543-586, 1933.

Cf. A354046.

A350028 Number of Euler tours of the complete graph on n vertices, minus a matching if n is even.

Original entry on oeis.org

1, 1, 2, 2, 264, 744, 129976320, 1847500800, 911520057021235200, 91507897551783002112, 257326999238092967427785160130560, 234051620220909442615820736748584960, 6705710151431658873046319662156165939200000000000000

Offset: 1

Günter Rote, Dec 08 2021

For even n, the graph is a cocktail party graph (cf. A297383). - Max Alekseyev, Jul 24 2025

			For n=6, if the edges 12,34,56 are removed from the complete graph and we fix the tour to start with the edge 13, we get 372 Euler tours. Here are the first and the last few in lexicographic order.
  1324152635461
  1324152645361
  1324153625461
  1324153645261
  1324154625361
  1324154635261
  1324162536451
  ...
  1364532516241
  1364532614251
  1364532615241.
This must be multiplied by 2 to account for the reversed tours, for a total of 744.

Eric Weisstein, Cocktail Party Graph, MathWorld.

Cf. A007082, A135388, A232545, A297383, A334554, A356366, A357855, A357856, A357857, A357885, A357886, A357887.

# for 3 <= n <= 9
def A(n,w="13"):
    if n%2==0 and len(w) > n*(n-1)//2 - n//2: return 2
    if n%2==1 and len(w) > n*(n-1)//2: return 2
    return sum(A(n, w+t) for t in "123456789"[:n]
        if t!=w[-1] and t+w[-1] not in w and w[-1]+t not in w
        and (n%2==1 or t+w[-1] not in "121 343 565 787"))

a(2n+1) = A135388(n) = A357887(2n+1,n(2n+1)) = A007082(n) * (n-1)!^(2*n+1); a(2n) = 2 * A297383(n) = A357887(2n,2n(n-1)) / (2n-1)!!. - Max Alekseyev, Oct 19 2022

a(1)-a(2) prepended, a(10)-a(13) added by Max Alekseyev, Jul 15 2025

A332637 The number of n X n replace matrices: binary matrices A where the i-th row contains exactly i zeros and A[i,j] >= A[j,i] for all i < j.

Original entry on oeis.org

1, 2, 8, 68, 1270, 53200, 5068960, 1109820882, 562711290616, 664773220895406

Offset: 1

Günter Rote, Feb 18 2020

Defined in Felsner, Definition 2.

			For n = 3, all nine 0-1-matrices with the correct number of zeros and ones in each row are replace matrices except
    [ 1 0 1 ]
A = [ 1 0 0 ]
    [ 0 0 0 ]

Stefan Felsner, On the number of arrangements of pseudolines, preprint.
Stefan Felsner, On the number of arrangements of pseudolines, Discrete & Computational Geometry, 18 (1997), 257-267.

According to [Felsner, Theorem 2] the number is at most 2^(0.6974*n^2) for large n.

a(8)-a(9) from Giovanni Resta, Feb 19 2020

a(10) from Giovanni Resta, Feb 21 2020

A325628 Number of mirror-symmetric Euclidean pseudo-order types: nondegenerate abstract order types of configurations of n points in the plane with a mirroring automorphism.

Original entry on oeis.org

0, 1, 1, 2, 3, 12, 28, 225, 825, 13103, 76188, 2358635, 21954947

Offset: 1

Manfred Scheucher and Günter Rote, Sep 07 2019

S. Felsner and J. E. Goodman, Pseudoline Arrangements. In: Toth, O'Rourke, Goodman (eds.) Handbook of Discrete and Computational Geometry, 3rd edn. CRC Press, 2018.

Cf. A006247, A325595.

Asymptotics: a(n) = 2^(Theta(n^2)). This is Bachmann-Landau notation, that is, there are constants n_0, c, and d, such that for every n >= n_0 the inequality 2^{c n^2} <= a(n) <= 2^{d n^2} is satisfied. For more information see e.g. the Handbook of Discrete and Computational Geometry. - Manfred Scheucher, Sep 12 2019

A325595 Number of symmetric Euclidean pseudo-order types: nondegenerate abstract order types of configurations of n points in the plane with a nontrivial automorphism.

Original entry on oeis.org

0, 1, 1, 2, 3, 12, 30, 230, 849, 13434, 76200, 2392066

Offset: 1

Manfred Scheucher and Günter Rote, Sep 07 2019

S. Felsner and J. E. Goodman, Pseudoline Arrangements. In: Toth, O'Rourke, Goodman (eds.) Handbook of Discrete and Computational Geometry, 3rd edn. CRC Press, 2018.

Cf. A006247, A325628.

Asymptotics: a(n) = 2^(Theta(n^2)). This is Bachmann-Landau notation, that is, there are constants n_0, c, and d, such that for every n >= n_0 the inequality 2^{c n^2} <= a(n) <= 2^{d n^2} is satisfied. For more information see e.g. the Handbook of Discrete and Computational Geometry. - Manfred Scheucher, Sep 12 2019

A306687 Triangular array read by rows: The sum of squares of the number of common points in all pairs of lattice paths from (0,0) to (x,y), for 0 <= y <= x (the unnormalized second moment).

Original entry on oeis.org

1, 4, 26, 9, 92, 474, 16, 240, 1704, 8084, 25, 520, 4879, 29560, 134450, 36, 994, 11928, 89928, 498140, 2208612, 49, 1736, 25956, 238440, 1580810, 8265432, 36024884, 64, 2832, 51648, 568128, 4442768, 27055808, 135873360, 584988840, 81, 4380, 95733, 1242648, 11320595, 79443000, 455434875, 2220096240, 9470766690

Offset: 0

Günter Rote, Mar 05 2019

			T(1,1) = 26, because the two lattice paths are DR and RD. (DR,DR) and (RD,RD) have three common points, (DR,RD) and (RD,DR) have two common points, and 2*3^2+2*2^2 = 26. - _Charlie Neder_, Jun 26 2019
The triangle begins:
   1,
   4,  26,
   9,  92,  474,
  16, 240, 1704,  8084,
  25, 520, 4879, 29560, 134450,
  ...

Lower triangle of the square array A324010.

a(x,y) = (x+y+1)*binomial(x+y+2,x+1)*binomial(x+y,x)-binomial(2*x+2*y+2,2*x+1)/2;
for (n=0, 10, for (k=0, n, print1(a(n,k), ", ")); print) \\ Michel Marcus, Apr 08 2019

T(x,y) = (x+y+1) * binomial(x+y+2,x+1) * binomial(x+y,x) - binomial(2*x+2*y+2,2*x+1) / 2.

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User: Günter Rote

A366409 Number of smooth convex lattice polygons with area n/2.

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A365641 The minimum number of ways to label each triangle of a triangulation of an n-gon with one of its vertices so that different triangles get different labels (minimum taken over all triangulations).

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Author

Keywords

Comments

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Python

A365620 Number of integer grid points on the circle around (0,0) with radius A088959(n).

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Author

Keywords

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Formula

A354046 Number of 4-dimensional chiral point groups of order n.

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A353341 Number of 4-dimensional point groups of order n.

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A350028 Number of Euler tours of the complete graph on n vertices, minus a matching if n is even.

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Python

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Extensions

A332637 The number of n X n replace matrices: binary matrices A where the i-th row contains exactly i zeros and A[i,j] >= A[j,i] for all i < j.

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Formula

Extensions

A325628 Number of mirror-symmetric Euclidean pseudo-order types: nondegenerate abstract order types of configurations of n points in the plane with a mirroring automorphism.

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Crossrefs

Formula

A325595 Number of symmetric Euclidean pseudo-order types: nondegenerate abstract order types of configurations of n points in the plane with a nontrivial automorphism.

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Author

Keywords

Links

Crossrefs

Formula

A306687 Triangular array read by rows: The sum of squares of the number of common points in all pairs of lattice paths from (0,0) to (x,y), for 0 <= y <= x (the unnormalized second moment).

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