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A333643 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around the arc of a semicircle. See Comments for precise definition. 2
1, 4, 11, 25, 50, 91, 154, 234, 375, 550, 769, 1079, 1456, 1783, 2500, 3196, 3987, 5016, 6175, 7348, 9086, 10879, 12836, 15250, 17875, 20682, 24129, 27811, 31419, 36425, 41416, 46664, 52921, 59500, 66489, 74481, 82954, 91807, 102050, 112750, 123700, 136654 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A semi-circular polygon with n+2 points is created by placing n+2 equally spaced vertices along a semicircle's arc, which includes the two end vertices. Now connect every pair of vertices by a straight line segment. The sequence gives the number of regions in the resulting figure.

Note that there is a curious relationship between the terms of this sequence and the number of regions in the 'general position' polygon given in A006522. They are a match except for every third term starting at a(8) = 234. Examining the images for n = 8,11,14,17 shows that these polygons have interior points at which three or more lines intersect, while the other n values have no such intersection points. Such multi-line intersection points will reduce the number of regions as compared to the general position polygon which has no multi-line intersection points. This is reflected by the terms in this sequence being lower than the corresponding value in A006522 for n = 8,11,14,... . Why every third value of n in this sequence starting at n = 8 leads to polygons having multiple line intersection points while other values of n do not is currently not known.

LINKS

Lars Blomberg, Table of n, a(n) for n = 1..80

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 = 7.

Scott R. Shannon, Illustration for n = 10.

Scott R. Shannon, Illustration for n = 12.

Scott R. Shannon, Illustration for n = 15.

Scott R. Shannon, Illustration for n = 17.

Scott R. Shannon, Illustration for n = 19.

Scott R. Shannon, Illustration for n = 20.

Scott R. Shannon, Illustration for n = 10 with random distance-based coloring.

Scott R. Shannon, Illustration for n = 15 with random distance-based coloring.

Scott R. Shannon, Illustration for n = 19 with random distance-based coloring.

Scott R. Shannon, Illustration for n = 20 with random distance-based coloring.

Wikipedia, Semicircle.

CROSSREFS

Cf. A333642, A333519, A007678, A290865, A092867, A331452, A331929, A331931.

Sequence in context: A051462 A006004 A290876 * A006522 A036837 A215052

Adjacent sequences:  A333640 A333641 A333642 * A333644 A333645 A333646

KEYWORD

nonn,more

AUTHOR

Scott R. Shannon and N. J. A. Sloane, Mar 31 2020

EXTENSIONS

More terms from Lars Blomberg, Apr 20 2020

STATUS

approved

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Last modified August 10 16:56 EDT 2020. Contains 336381 sequences. (Running on oeis4.)