A234351 -id:A234351 - cp's OEIS frontend

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.

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

Offset: 1

Heinrich Ludwig, Dec 24 2013

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.

			Triangle begins
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;
...
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 . .

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

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.

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.

Original entry on oeis.org

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

Offset: 3

Heinrich Ludwig, Dec 30 2013

Column 4 of triangle A234350.

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

			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

Cf. A234350, A194132
Arranging 3 points: A234351
Arranging 5 points: A234353
Arranging 6 points: A234354

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

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.

Original entry on oeis.org

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

Offset: 4

Heinrich Ludwig, Dec 30 2013

Column 5 of triangle A234350.

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

			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

Cf. A234350, A194133
Arranging 3 points: A234351
Arranging 4 points: A234352
Arranging 6 points: A234354

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

A234354 Number of non-equivalent (mod D_3) ways to arrange 6 indistinguishable points on a triangular grid of side n so that no three points are collinear.

Original entry on oeis.org

1, 47, 1112, 11743, 82988, 433372, 1820163, 6459496, 20287372, 57129426, 147915285, 356109110, 804785159, 1722171831

Offset: 4

Heinrich Ludwig, Dec 30 2013

Column 6 of triangle A234350.

Without the restriction "non-equivalent (mod D_3)" the numbers are given by A194134.

			There is a(4) = 1 non-equivalent way to arrange 6 indistinguishable points (x) on a triangular grid of side 4:
     .
    x x
   x . x
  . x x .

Cf. A234350, A194134
Arranging 3 points: A234351
Arranging 4 points: A234352
Arranging 5 points: A234353

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

cp's OEIS Frontend

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.

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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.

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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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A234354 Number of non-equivalent (mod D_3) ways to arrange 6 indistinguishable points on a triangular grid of side n so that no three points are collinear.

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Author

Keywords

Comments

Examples

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Extensions