

A345459


Number of polygons formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.


8



0, 4, 80, 568, 2024, 6052, 12144, 26976, 45024, 76724, 116840, 191128, 245976, 388452, 501888, 661476, 870168, 1199724, 1402096, 1911384, 2188320, 2739280, 3371264, 4224288, 4617224, 5801372, 6780568
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OFFSET

0,2


COMMENTS

The width/height of the entire figure grows as ~ 2*n^3 for large n. See the Formula section below.


LINKS

Table of n, a(n) for n=0..26.
Scott R. Shannon, Image for n = 2. In this and other images the square's points are highlighted as white dots while the outer open regions, which are not counted, are darkened. The key for the edgenumber coloring is shown at the topleft of the image.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.


FORMULA

a(n) = A345650(n)  A345649(n) + 1.
Assuming the square is of size n x n centered on the origin the x (or y) offset for the eight outermost vertices is n^3  2*n^2 + 3*n/2, which have a corresponding y (or x) offset of n^2  3*n/2 + 1. The total distance from the origin of these vertices is sqrt(n^6  4*n^5 + 8*n^4  9*n^3 + 13*n^2/2  3*n + 1).


EXAMPLE

a(2) = 80. Connecting the 8 perimeter points results in the creation of fortyeight 3gons and eight 4gons inside the square while creating twentyfour 3gons outside the square, giving eighty polygons in total. See the linked images.


CROSSREFS

Cf. A255011 (number inside the square), A345648 (number outside the square), A345649 (number of vertices), A345650 (number of edges), A344993, A344857, A092098, A007678.
Sequence in context: A192790 A211152 A093854 * A269146 A192834 A054322
Adjacent sequences: A345456 A345457 A345458 * A345460 A345461 A345462


KEYWORD

nonn,more


AUTHOR

Scott R. Shannon and N. J. A. Sloane, Jun 20 2021


STATUS

approved



