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

1,1

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.

Links

Table of n, a(n) for n=1..27.

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.

Example

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

Crossrefs

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

Sequence in context: A211150 A281534 A006415 * A181396 A354063 A196979

Adjacent sequences: A333431 A333432 A333433 * A333435 A333436 A333437

Keyword

nonn,more

Author

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

Extensions

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

Status

approved