A333460 -id:A333460 - cp's OEIS frontend

A333434 The number of regions inside a diagonal-edged (or diamond-shaped) checkerboard of width and height 2n-1 formed by the straight line segments mutually connecting any two of the 8n-4 vertices on the perimeter.

4, 104, 1080, 5220, 15508, 39088, 81464, 144292, 261544, 415552, 610460, 942032, 1303848, 1803360, 2461232, 3250284, 4182552, 5269080, 6818764, 8326188, 10336548, 12621292, 14882600, 18368708, 21377496, 25168908, 29994204

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 21 2020

The diagonal-edged checker board of width and height 2*n-1 contains 8*n-4 vertices lying on a 2D square grid as shown in the examples below. Join every pair of vertices by a line segment, provided the line does not extend beyond the boundary of the board. The sequence gives the number of regions in the resulting figure.

			For n = 1 the board is a single square with 4 vertices on the corners.
For n = 2 the board contains 12 vertices, represented by '*', shown below:
          *---*
          |   |
      *---*   *---*
      |           |
      *---*   *---*
          |   |
          *---*
.
For n = 3 the board contains 20 vertices, represented by '*', shown below:
          *---*
          |   |
      *---*   *---*
      |           |
  *---*           *---*
  |                   |
  *---*           *---*
      |           |
      *---*   *---*
          |   |
          *---*
.

Scott R. Shannon, Illustration for n = 2.
Scott R. Shannon, Illustration for n = 3.
Scott R. Shannon, Illustration for n = 4.
Scott R. Shannon, Illustration for n = 5.
Scott R. Shannon, Illustration for n = 6.
Scott R. Shannon, Illustration for n = 2 using random distance-based coloring.
Scott R. Shannon, Illustration for n = 3 using random distance-based coloring.
Scott R. Shannon, Illustration for n = 4 using random distance-based coloring.
Scott R. Shannon, Illustration for n = 5 using random distance-based coloring.
Scott R. Shannon, Illustration for n = 6 using random distance-based coloring.

Cf. A333458 (n-gons), A333459 (vertices), A333460 (edges), A331452, A331456, A331911.

a(8)-a(27) from Lars Blomberg, Jun 03 2020

A333458 Irregular table read by rows: Take a diagonal-edged (or diamond-shaped) checkerboard with all diagonals drawn, as in A333434. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

Original entry on oeis.org

4, 84, 20, 692, 312, 72, 4, 2708, 2020, 400, 88, 0, 4, 7876, 5764, 1580, 264, 20, 4, 18220, 14868, 4960, 904, 112, 24, 36764, 30624, 11224, 2496, 304, 52, 66004, 55192, 18624, 3952, 472, 48, 111764, 97840, 40008, 9912, 1740, 244, 36, 175156, 158280, 63632, 15744, 2560, 156, 8, 16

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 22 2020

See A333434 for the definition and images of the checkerboard.

			Table begins:
4;
84, 20;
692, 312, 72, 4;
2708, 2020, 400, 88, 0, 4;
7876, 5764, 1580, 264, 20, 4;
18220, 14868, 4960, 904, 112, 24;
36764, 30624, 11224, 2496, 304, 52;
66004, 55192, 18624, 3952, 472, 48;
111764, 97840, 40008, 9912, 1740, 244, 36;
175156, 158280, 63632, 15744, 2560, 156, 8, 16;
...

Lars Blomberg, Table of n, a(n) for n = 1..201 (the first 27 rows)

Cf. A333434 (regions), A333459 (vertices), A333460 (edges), A331911, A331451, A333025, A333037.

a(32) and beyond from Lars Blomberg, Jun 03 2020

A333459 Number of vertices in an diagonal-edged (or diamond-shaped) checkerboard with width and height 2*n-1 (see Comments in A333434 for definition).

Original entry on oeis.org

5, 69, 785, 4177, 12681, 33601, 71777, 125389, 238801, 379793, 548293, 878397, 1203101, 1673693, 2311477, 3063609, 3934365, 4931077, 6501545, 7873689, 9852941, 12071685, 14067849, 17728005, 20417893, 24047817, 29003717

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 22 2020

See A333434 for the definition and images of the checkerboard.

Cf. A333434 (regions), A333458 (n-gons), A333460 (edges), A333035, A333031, A330850.

a(8)-a(27) from Lars Blomberg, Jun 03 2020

cp's OEIS Frontend

A333434 The number of regions inside a diagonal-edged (or diamond-shaped) checkerboard of width and height 2n-1 formed by the straight line segments mutually connecting any two of the 8n-4 vertices on the perimeter.

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A333458 Irregular table read by rows: Take a diagonal-edged (or diamond-shaped) checkerboard with all diagonals drawn, as in A333434. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

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A333459 Number of vertices in an diagonal-edged (or diamond-shaped) checkerboard with width and height 2*n-1 (see Comments in A333434 for definition).

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Author

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cp's OEIS Frontend

A333434 The number of regions inside a diagonal-edged (or diamond-shaped) checkerboard of width and height 2*n-1 formed by the straight line segments mutually connecting any two of the 8*n-4 vertices on the perimeter.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A333458 Irregular table read by rows: Take a diagonal-edged (or diamond-shaped) checkerboard with all diagonals drawn, as in A333434. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A333459 Number of vertices in an diagonal-edged (or diamond-shaped) checkerboard with width and height 2*n-1 (see Comments in A333434 for definition).

Views

Author

Keywords

Comments

Crossrefs

Extensions

A333434 The number of regions inside a diagonal-edged (or diamond-shaped) checkerboard of width and height 2n-1 formed by the straight line segments mutually connecting any two of the 8n-4 vertices on the perimeter.