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.

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

Author

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

Keywords

• nonn
• more

Comments

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.

Examples

			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:
          *---*
          |   |
      *---*   *---*
      |           |
  *---*           *---*
  |                   |
  *---*           *---*
      |           |
      *---*   *---*
          |   |
          *---*
.
		

Links

• 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.

Crossrefs

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

Extensions

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