cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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.

Original entry on oeis.org

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

Views

Author

Heinrich Ludwig, Dec 24 2013

Keywords

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.

Examples

			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 . .

Links

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.

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.

Original entry on oeis.org

1, 5, 22, 77, 217, 530, 1149, 2288, 4231, 7400, 12322, 19725, 30486, 45768, 66917, 95669
Offset: 2

Views

Author

Heinrich Ludwig, Dec 30 2013

Keywords

Comments

Column 3 of triangle A234350.
Without the restriction "non-equivalent (mod D_3)" the numbers are given by A194131.

Examples

			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

Extensions

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

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

Views

Author

Heinrich Ludwig, Dec 30 2013

Keywords

Comments

Column 4 of triangle A234350.
Without the restriction "non-equivalent (mod D_3)" the numbers are given by A194132.

Examples

			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

Extensions

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

Views

Author

Heinrich Ludwig, Dec 30 2013

Keywords

Comments

Column 6 of triangle A234350.
Without the restriction "non-equivalent (mod D_3)" the numbers are given by A194134.

Examples

			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 .

Crossrefs

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

Extensions

a(14)-a(17) from Heinrich Ludwig, Apr 19 2016
Showing 1-4 of 4 results.

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