A331455 Number of regions in a "cross" of width 3 and height n (see Comments for definition).

64, 104, 176, 304, 492, 778, 1176, 1732, 2446, 3416, 4614, 6172, 8060, 10340, 13052, 16388, 20228, 24852, 30134, 36206, 43076, 51092, 60010, 70186, 81498, 94180, 108140, 123938, 141074, 160308, 181320, 204328, 229288, 256574, 285856, 318124, 352838, 390338

Offset

2,1

Comments

This "cross" of height n consists of a vertical column of n >= 2 squares with two additional squares extending to the left and right of the second square. (See illustrations.)

There are n+2 squares in all. The number of vertices is 3*n+2.

Now join every pair of vertices by a line segment, provided the line does not extend beyond the boundary of the cross. The sequence gives the number of regions in the resulting figure.

Links

Lars Blomberg, Table of n, a(n) for n = 2..50

Scott R. Shannon, Illustration for cross of height 2.

Scott R. Shannon, Illustration for cross of height 3.

Scott R. Shannon, Illustration for cross of height 4.

Scott R. Shannon, Illustration for cross of height 5.

Scott R. Shannon, Illustration for cross of height 6.

Scott R. Shannon, Illustration for cross of height 9.

Scott R. Shannon, Illustration for cross of height 3 using random distance-based coloring.

Scott R. Shannon, Illustration for cross of height 4 using random distance-based coloring.

Scott R. Shannon, Illustration for cross of height 5 using random distance-based coloring.

Scott R. Shannon, Illustration for cross of height 6 using random distance-based coloring.

Scott R. Shannon, Illustration for cross of height 7 using random distance-based coloring.

Scott R. Shannon, Colored illustration for a different-shaped cross, with arms of lengths 2,2,4. (There are 21858 regions.)

N. J. A. Sloane, Illustration for cross of height 2.

N. J. A. Sloane, Illustration for cross of height 3. (One of the "arms" has been cropped by the scanner, but all four arms are the same.)

N. J. A. Sloane, Illustration for cross of height 4.

N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.

Crossrefs

Cf. A330848 (n-gons), A330850 (vertices), A330851 (edges).

See A331456 for crosses in which the arms have equal length.

A331452 is a similar sequence for a rectangular region; A007678 for a polygonal region.

Sequence in context: A316419 A306165 A317382 * A039481 A094682 A118158

Adjacent sequences: A331452 A331453 A331454 * A331456 A331457 A331458

Keyword

nonn

Author

Scott R. Shannon and N. J. A. Sloane, Jan 28 2020

Extensions

a(11) and beyond from Lars Blomberg, May 31 2020

Status

approved