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

Author

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

Keywords

• nonn

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.

Extensions

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