A018808 -id:A018808 - cp's OEIS frontend

A333284 Triangle read by rows: T(m,n) (m >= n >= 1) = number of vertices formed by drawing the line segments connecting any two of the (m+1) X (n+1) lattice points in an m X n grid of squares and extending them to the boundary of the grid.

5, 13, 37, 35, 129, 405, 75, 289, 933, 2225, 159, 663, 2155, 5157, 11641, 275, 1163, 3793, 9051, 20341, 35677, 477, 2069, 6771, 16129, 36173, 63987, 114409, 755, 3251, 10727, 25635, 57759, 102845, 183961

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 16 2020

If we only joined pairs of the 2(m+n) boundary points, we would get A331453. If we did not extend the lines to the boundary of the grid, we would get A288180. (One of the links below shows the difference between the three definitions in the case m=3, n=2.)

See A333282 for a large number of colored illustrations.

			Triangle begins:
5,
13, 37,
35, 129, 405,
75, 289, 933, 2225,
159, 663, 2155, 5157, 11641,
275, 1163, 3793, 9051, 20341, 35677,
477, 2069, 6771, 16129, 36173, 63987, 114409,
755, 3251, 10727, 25635, 57759, 102845, 183961, ...
...
T(7,7) corrected Mar 19 2020

Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.
Seppo Mustonen, Statistical accuracy of geometric constructions, 2008.
Seppo Mustonen, Statistical accuracy of geometric constructions, 2008 [Local copy]
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points, 2009
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points, 2009 [Local copy]
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points, 2010
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points, 2010 [Local copy]
N. J. A. Sloane, Illustration of T(3,2) = 129. [Black lines correspond to A331453(3,2), black + red lines correspond to A288180(3,2), and black + red + blue lines to T(3,2)]
N. J. A. Sloane, Illustration of T(3,3) = 405 [Black lines correspond to A288180(3,3), and black + red lines to T(3,3)]

Cf. A288187, A331452, A288180, A331453, A333282 (regions), A333283 (edges). Column 1 is A331755. The main diagonal is A333285.

More terms and corrections from Scott R. Shannon, Mar 21 2020

A241220 Number of lines through at least two points of a centered hexagonal grid of size n.

Original entry on oeis.org

0, 15, 84, 312, 852, 1878, 3654, 6546, 10680, 16668, 25002, 35910, 50136, 68190, 90462, 118200, 152274, 192828, 240480, 296880, 361962, 437832, 525756, 625440, 739146, 867864, 1011822, 1174062, 1354572, 1554114, 1775568, 2020848, 2289054, 2582760, 2905410

Offset: 1

Martin Renner, Apr 17 2014

nonn

A centered hexagonal grid of size n is a grid with A003215(n-1) points forming a hexagonal lattice.

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Hex Number.

Cf. A018808, A241219, A003215.

c(n,s,fmin,fmax)={sum(k=1+s, n, max(0, fmax(k-s)-max(fmin(k)-1,if(k-2*s>0,fmax(k-2*s)))))}
b(n, u, v)={c(2*n-1, u, i->max(0,i-n)+1+i\u*v, i->min(i,n)+n-1+i\u*v)}
a(n)={3*((n>1)*(2*n-1) + sum(u=1, 2*n-3, sum(v=1, 2*n-2-u, if(gcd(u,v)==1, b(n,u,v), 0))))} \\ Andrew Howroyd, Sep 18 2017

a(15)-a(16) from Martin Renner, Apr 27 2014

Terms a(17) and beyond from Andrew Howroyd, Sep 18 2017

A119438 Number of sets of points determined by the intersection of a line with an n X n grid of points.

Original entry on oeis.org

1, 2, 11, 30, 79, 166, 343, 586, 1003, 1574, 2407, 3418, 4867, 6630, 9027, 11794, 15203, 19190, 24251, 29906, 36911, 44830, 54071, 64178, 76251, 89574, 105051, 121762, 140751, 161478, 185367, 210906, 240075, 271678, 306635, 343706, 384667

Offset: 0

Franklin T. Adams-Watters, May 19 2006

nonn

Each singleton point is determined by all but finitely many of the family of lines passing through that point and the empty set is determined by any randomly positioned line.

Cf. A018808, A119439.

a(n) = A018808(n) + n^2 + 1.

A234566 1/a(n) is the area of the smallest triangle delimited by 3 lines each passing through at least 2 points of an n X n unitary spaced grid.

Original entry on oeis.org

4, 30, 770, 5148, 30566, 89900, 219960, 614460, 1146596, 2624076, 4299916, 8432732, 11016390, 22391148, 28183214

Offset: 2

Giovanni Resta, Dec 28 2013

Only non-degenerate triangles are considered.
Conjecture: sequence is well-defined, i.e., a(n) is an integer for every n > 1.
For n > 1, n odd, a(n) >= 4(n-2)^2(n^2-3n+1)(n^2-3n+3), with equality for  n = 5,7,...,15. The bound is obtained considering the 3 lines passing by the 3 pairs of points {(0,1), (n-1, n-1)}, {(n-2, 0), (1, n-2)}, and {((n-1)/2, n-1), ((n-3)/2, 0)}.
We may consider the similar problem of finding the largest triangle. Here, the areas for n>=2 are 1/2, 9/2, 25, 100, 289, 676, 1369,... so it appears that for n >= 4 the maximal area is ((n-2)^2+1)^2, (cf. A082044) obtained via the lines passing through the points {(0,2), (1,n-1)}, {(n-2,0), (n-1,n-2)}, and {(0,0), (1,n-1)}.

			For n=2, consider the 2 X 2 grid formed by the points with coordinates (0,0), (0,1), (1,0) and (1,1). The two diagonals and the line passing through (0,0) and (1,0) form a triangle whose area is 1/4 and since no smaller triangle can be formed in this way, a(2) = 4.

Giovanni Resta, Illustration of the first 9 terms

Cf. A018808, A082044.

For n>1 odd, a(n) >= 4(n-2)^2 (n^2-3n+1)(n^2-3n+3).

A331780 One-half of the number of lines through at least 2 points of an n X n grid of points.

Original entry on oeis.org

0, 0, 3, 10, 31, 70, 153, 268, 469, 746, 1153, 1648, 2361, 3230, 4415, 5784, 7473, 9450, 11963, 14772, 18255, 22194, 26793, 31824, 37837, 44474, 52187, 60516, 69983, 80318, 92233, 104972, 119525, 135294, 152739, 171240, 191685, 213510, 237915, 263640

Offset: 0

N. J. A. Sloane, Feb 10 2020

nonn

M. A. Alekseyev, M. Basova, N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM J. Disc. Math. 29(1), 2015, pp. 157-165.

Equals A018808/2.

A333285 The main diagonal of the triangular array A333284.

Original entry on oeis.org

5, 37, 405, 2225, 11641, 35677, 114409, 295701, 718469, 1475709, 3093025, 5771929, 10895273, 18785841, 31414269, 50274501, 81288641, 124066161, 190860537, 282399889, 411505049, 580614301, 824814797, 1138709849, 1570665877, 2115178249, 2833746309, 3732420861, 4937226173

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 16 2020

nonn

See A333282, A333283, and A333284 for further information, illustrations, etc.

Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.
Seppo Mustonen, Statistical accuracy of geometric constructions, September 02, 2008.
Seppo Mustonen, Statistical accuracy of geometric constructions, 2008 [Local copy]
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points, 2009
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points, 2009 [Local copy]
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points, 2010
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points, 2010 [Local copy]

Cf. A333282, A333283, A333284.

cp's OEIS Frontend

A333284 Triangle read by rows: T(m,n) (m >= n >= 1) = number of vertices formed by drawing the line segments connecting any two of the (m+1) X (n+1) lattice points in an m X n grid of squares and extending them to the boundary of the grid.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A241220 Number of lines through at least two points of a centered hexagonal grid of size n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A119438 Number of sets of points determined by the intersection of a line with an n X n grid of points.

Views

Author

Keywords

Comments

Crossrefs

Formula

A234566 1/a(n) is the area of the smallest triangle delimited by 3 lines each passing through at least 2 points of an n X n unitary spaced grid.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A331780 One-half of the number of lines through at least 2 points of an n X n grid of points.

Views

Author

Keywords

Links

Crossrefs

A333285 The main diagonal of the triangular array A333284.

Views

Author

Keywords

Comments

Links

Crossrefs