cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A345459 Number of polygons formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

Original entry on oeis.org

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
Offset: 0

Views

Author

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

Keywords

Comments

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

Examples

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

Links

  • 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 edge-number coloring is shown at the top-left 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.

Crossrefs

Cf. A255011 (number inside the square), A345648 (number outside the square), A345649 (number of vertices), A345650 (number of edges), A344993, A344857, A092098, A007678.

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).

A347750 Number of intersection points when every pair of vertices of a row of n adjacent congruent rectangles are joined by an infinite line.

Original entry on oeis.org

0, 5, 17, 57, 133, 297, 525, 925, 1477, 2289, 3277, 4701, 6437, 8805, 11541, 14917, 18869, 23893, 29509, 36473, 44349, 53545, 63605, 75629, 88901, 104325, 120981, 139913, 160581, 184409, 209885, 238989, 270525, 305413, 342413, 383301, 426949, 475757, 527205, 583261, 642821, 708717, 777829
Offset: 0

Views

Author

Scott R. Shannon and N. J. A. Sloane, Sep 12 2021

Keywords

Examples

			a(1) = 5 as connecting the four vertices of a single rectangle forms one new vertex inside the rectangle, giving a total of 4 + 1 = 5 total intersection points.
a(2) = 17 as connecting the six vertices of two adjacent rectangles forms seven vertices inside the rectangles while also forming four vertices outside the rectangles. The total number of intersection points is then 6 + 7 + 4 = 17.
See the linked images for further examples.

Links

Crossrefs

Cf. A344993 (number of polygons), A347751 (number of edges), A159065, A331755, A092275 (number of intersections resp. inside the rectangles, on or inside them, above them).

Formula

a(n) = A347751(n) - A344993(n) + 1.
It seems that a(n) = 2 * A159065(n+1) + 3 for n>0. - Andrei Zabolotskii, Jul 03 2025

A345649 Number of vertices formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

Original entry on oeis.org

0, 5, 61, 465, 1585, 5257, 9749, 24025, 38381, 67177, 100889, 176005, 210033, 360877, 450349, 589581, 779541, 1127509, 1251805, 1806061, 1970129, 2504401, 3116945, 4017701, 4163753, 5433657, 6335589
Offset: 0

Views

Author

Scott R. Shannon and N. J. A. Sloane, Jun 21 2021

Keywords

Comments

See A345459 for images of the polygons.

Crossrefs

Cf. A345459 (number of polygons), A345650 (number of edges), A255011, A345648, A344993, A344857, A092098, A007678.

Formula

a(n) = A345650(n) - A345459(n) + 1.

A345650 Number of polygon edges formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

Original entry on oeis.org

0, 8, 140, 1032, 3608, 11308, 21892, 51000, 83404, 143900, 217728, 367132, 456008, 749328, 952236, 1251056, 1649708, 2327232, 2653900, 3717444, 4158448, 5243680, 6488208, 8241988, 8780976, 11235028, 13116156
Offset: 0

Views

Author

Scott R. Shannon and N. J. A. Sloane, Jun 21 2021

Keywords

Comments

See A345459 for images of the polygons.

Crossrefs

Cf. A331448 (number inside the square), A345459 (number of polygons), A345649 (number of vertices), A255011, A345648, A344993, A344857, A092098, A007678.

Formula

a(n) = A345459(n) + A345649(n) - 1.

A386560 Place a point on the integer coordinates, up to |n|, along all four axial directions on a Cartesian plane, and then join an infinite straight line between every pair of points: a(n) is the number of finite regions created in the resulting graph.

Original entry on oeis.org

4, 72, 424, 1396, 3536, 7292, 14272, 24332, 39356, 59920, 91348, 128084, 182664, 245804, 323116, 418552, 547820, 684680, 869388, 1060892, 1289564, 1560920
Offset: 1

Views

Author

Scott R. Shannon, Jul 26 2025

Keywords

Links

Crossrefs

Cf. A386559 (vertices), A386561 (edges), A386562 (k-gons), A344993, A344857, A344279, A345459.

Formula

a(n) = A386561(n) - A386559(n) + 1 by Euler's formula.

A345648 Number of polygons formed outside an n X n square when connecting all 4n points on the perimeter of the square by infinite lines.

Original entry on oeis.org

0, 0, 24, 228, 904, 2788, 5880, 13008, 22120, 37976, 58584, 95472, 125016, 195816, 255064, 337916, 444760, 611760, 719800, 978388, 1127088, 1411756, 1736776, 2174584, 2389552, 2995336, 3504768
Offset: 0

Views

Author

Scott R. Shannon and N. J. A. Sloane, Jun 21 2021

Keywords

Comments

For n = 0 to n = 11 the number of polygons formed outside the n x n square is less than the number formed inside the square. This changes for n >= 12 when the number formed outside becomes greater. The ratio of the number of polygons outside to the number inside for n = 26 is about 1.07 . If this ratio is unbounded or approaches some finite value as n -> infinity is not known.
See A345459 for images of the polygons.

Crossrefs

Cf. A345459 (number inside and outside with square), A255011 (number inside the square), A344993, A344857, A092098, A007678.

Formula

a(n) = A345459(n) - A255011(n).

A347751 Number of finite edges in the graph formed when every pair of vertices of a row of n adjacent congruent rectangles are joined by an infinite line.

Original entry on oeis.org

0, 8, 36, 124, 300, 664, 1200, 2108, 3388, 5232, 7568, 10852, 14892, 20288, 26704, 34540, 43812, 55400, 68584, 84684, 103004, 124216, 147888, 175820, 206788, 242424, 281560, 325708, 374148, 429416, 489000, 556412, 629804, 710536, 797280, 892564, 994588, 1107744, 1228432, 1359292, 1498788
Offset: 0

Views

Author

Scott R. Shannon and N. J. A. Sloane, Sep 12 2021

Keywords

Comments

See A344993 and A347750 for images of the rectangles.

Examples

			a(1) = 8 as connecting the four vertices of a single rectangle forms four new edges inside the rectangle, giving a total of 4 + 4 = 8 total edges.
a(2) = 36 as connecting the six vertices of two adjacent rectangles forms twenty-two edges inside the rectangles while also forming eight edges outside the rectangles. The total number of edges is then 6 + 22 + 8 = 36.

Crossrefs

Cf. A344993 (number of polygons), A347750 (number of intersections), A331757 (number of edges on or inside the rectangles).

Formula

a(n) = A344993(n) + A347750(n) - 1.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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