A234354 -id:A234354

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A234350

Triangle T(n, k) = Number of non-equivalent (mod D_3) ways to arrange k indistinguishable points on a triangular grid of side n so that no three points are collinear. Triangle read by rows.

+10
6

1, 1, 1, 1, 2, 4, 5, 2, 3, 10, 22, 24, 8, 1, 4, 22, 77, 153, 140, 47, 2, 5, 41, 217, 713, 1290, 1112, 322, 15, 7, 72, 530, 2557, 7374, 11743, 8783, 2412, 143, 1, 8, 116, 1149, 7661, 32477, 82988, 116154, 77690, 19621, 1220, 5, 10, 180, 2288, 20055, 116420, 433372

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

The triangle T(n, k) is irregularly shaped: 1 <= k <= A234349(n). First row corresponds to n = 1.

The maximal number of points that can be placed on a triangular grid of side n so that no three points are collinear is given by A234349(n).

Without the restriction "non-equivalent (mod D_3)" the numbers are given by A194136.

LINKS

Heinrich Ludwig, Table of n, a(n) for n = 1..152

EXAMPLE

Triangle begins

1, 1, 1;

2, 4, 5, 2;

3, 10, 22, 24, 8, 1;

4, 22, 77, 153, 140, 47, 2;

5, 41, 217, 713, 1290, 1112, 322, 15;

7, 72, 530, 2557, 7374, 11743, 8783, 2412, 143, 1;

8, 116, 1149, 7661, 32477, 82988, 116154, 77690, 19621, 1220, 5;

...

There are e.g. T(8, 11) = 5 non-equivalent ways to arrange 11 indistinguishable points (X) on a triangular grid of side 8 so that no point triple is collinear. As examples of the 5 solutions the 2 symmetrical ones are shown.

. .

. . . .

. X . . X .

X . . X X . . X

X . . . X . X . X .

. . X X . . X . . . . X

. X . . . X . . . X . X . .

. . X . . X . . . . X . . X . .

CROSSREFS

Cf. A194136

Row lengths are given by A234349

Column 1 is A001399

Column 2 is A227327 for n >= 2

Column 3 is A234351

Column 4 is A234352

Column 5 is A234353

Column 6 is A234354.

KEYWORD

nonn,tabf,nice

AUTHOR

Heinrich Ludwig, Dec 24 2013

STATUS

approved

A234351

Number of non-equivalent (mod D_3) ways to arrange 3 indistinguishable points on a triangular grid of side n so that no three points are collinear.

+10
4

1, 5, 22, 77, 217, 530, 1149, 2288, 4231, 7400, 12322, 19725, 30486, 45768, 66917, 95669

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,2

COMMENTS

Column 3 of triangle A234350.

Without the restriction "non-equivalent (mod D_3)" the numbers are given by A194131.

LINKS

Table of n, a(n) for n=2..17.

EXAMPLE

There are a(3) = 5 non-equivalent ways to arrange 3 indistinguishable points (x) on a triangular grid of side 3:

x . x x x

. . x x x x . . x .

x . x . x . . . . . x x . x .

CROSSREFS

Cf. A234350, A194131

Arranging 4 points: A234352

Arranging 5 points: A234353

Arranging 6 points: A234354

KEYWORD

nonn,more

AUTHOR

Heinrich Ludwig, Dec 30 2013

EXTENSIONS

a(14)-a(17) from Heinrich Ludwig, Apr 19 2016

STATUS

approved

A234352

Number of non-equivalent (mod D_3) ways to arrange 4 indistinguishable points on a triangular grid of side n so that no three points are collinear.

+10
4

2, 24, 153, 713, 2557, 7661, 20055, 47153, 101821, 205322, 390580, 707708, 1229969, 2060777, 3344487

(list; graph; refs; listen; history; text; internal format)

OFFSET

3,1

COMMENTS

Column 4 of triangle A234350.

Without the restriction "non-equivalent (mod D_3)" the numbers are given by A194132.

LINKS

Table of n, a(n) for n=3..17.

EXAMPLE

There are a(3) = 2 non-equivalent ways to arrange 4 indistinguishable points (x) on a triangular grid of side 3:

x .

x x x x

. x . x . x

CROSSREFS

Cf. A234350, A194132

Arranging 3 points: A234351

Arranging 5 points: A234353

Arranging 6 points: A234354

KEYWORD

nonn,more

AUTHOR

Heinrich Ludwig, Dec 30 2013

EXTENSIONS

a(14)-a(17) from Heinrich Ludwig, Apr 19 2016

STATUS

approved

A234353

Number of non-equivalent (mod D_3) ways to arrange 5 indistinguishable points on a triangular grid of side n so that no three points are collinear.

+10
4

8, 140, 1290, 7374, 32477, 116420, 358112, 976063, 2425756, 5560046, 11946923, 24274195, 46975815, 87140153

(list; graph; refs; listen; history; text; internal format)

OFFSET

4,1

COMMENTS

Column 5 of triangle A234350.

Without the restriction "non-equivalent (mod D_3)" the numbers are given by A194133.

LINKS

Table of n, a(n) for n=4..17.

EXAMPLE

There are a(4) = 8 non-equivalent ways to arrange 5 indistinguishable points (x) on a triangular grid of side 4. Some examples:

x x . . .

x x . . x . x x x .

. . . x . x x . x . x . x . x

. x x . . x x . . x x . x . . x . x . x

CROSSREFS

Cf. A234350, A194133

Arranging 3 points: A234351

Arranging 4 points: A234352

Arranging 6 points: A234354

KEYWORD

nonn,more

AUTHOR

Heinrich Ludwig, Dec 30 2013

EXTENSIONS

a(14)-a(17) from Heinrich Ludwig, Apr 19 2016

STATUS

approved

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