A331456 Number of regions in an equal-armed cross with arms of length n (see Comments for definition).

4, 104, 568, 1900, 4808, 10180, 19180, 33132, 53628, 82432, 121448, 172948, 239356, 323168, 427272, 554892, 709476, 893772, 1111588, 1367292, 1664604, 2008240, 2402560, 2852532, 3363280, 3938712, 4585568, 5308720, 6112736, 7006068, 7994412, 9084788, 10281812

Offset: 0

Author

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

Keywords

• nonn

Comments

This cross of height n consists of a central square with 4 arms of length n.

There are 4n+1 squares in all. The number of vertices is 8n+4.

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.

See A337641 for information about these regions, their numbers of sides, their coordinates, and for further illustrations. - N. J. A. Sloane, Sep 17 2020

Links

• Lars Blomberg, Table of n, a(n) for n = 0..48
• Scott R. Shannon, Colored illustration for a(0).
• Scott R. Shannon, Colored illustration for a(1).
• Scott R. Shannon, Colored illustration for a(2).
• Scott R. Shannon, Colored illustration for a(3).
• Scott R. Shannon, Colored illustration for a(4).
• Scott R. Shannon, Colored illustration for a(5).
• Scott R. Shannon, Colored illustration for a(9).
• Scott R. Shannon, Colored illustration for a(1) classifying nodes and cells.
• Scott R. Shannon, Colored illustration for a(2) classifying nodes and cells.
• Scott R. Shannon, Colored illustration for a(3) classifying nodes and cells.
• Scott R. Shannon, Colored illustration for a(4) classifying nodes and cells.
• Scott R. Shannon, Colored illustration for a(5) classifying nodes and cells.
• Scott R. Shannon, Colored illustration for a(6) classifying nodes and cells.
• N. J. A. Sloane, Illustration for a(1). (One of the "arms" has been cropped by the scanner, but all four arms are the same.)
• 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. A333035 (vertices), A333036 (edges), A333037 (n-gons), A337641.

See A331455 for a different family of crosses.

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

Cf. A331458.

Extensions

a(11) and beyond from Lars Blomberg, May 30 2020