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 1-8 of 8 results.

A344279 Number of polygons formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.

Original entry on oeis.org

1, 12, 102, 396, 1198, 2748, 5539, 10272, 16986, 26934, 41179, 60804, 84769, 119022, 157947, 206352, 268030, 347430, 432820, 543210, 659238, 801804, 970429, 1171662, 1371040, 1627398, 1904550, 2213712, 2555320, 2971260, 3373399, 3881838, 4399329, 4988502, 5610543, 6315312
Offset: 1

Views

Author

Scott R. Shannon, Jun 22 2021

Keywords

Links

Crossrefs

Cf. A344657 (number of vertices), A344896 (number of edges), A346446 (number of k-gons), A092867 (number polygons inside the triangle), A343755 (number of regions), A345459, A344857.

Formula

a(n) = A344896(n) - A344657(n) + 1.

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).

A358556 Triangle read by rows: T(n,k) is the number of regions formed when n points are placed along each edge of a square that divide the edges into n+1 equal parts and a line is continuously drawn from the current point to that k points, 2 <= k <= 2*n, counterclockwise around the square until the starting point is again reached.

Original entry on oeis.org

2, 5, 21, 2, 5, 5, 4, 61, 2, 5, 29, 5, 73, 25, 105, 2, 5, 25, 5, 5, 31, 141, 11, 157, 2, 5, 5, 5, 85, 5, 153, 4, 25, 61, 229, 2, 5, 25, 5, 73, 33, 5, 15, 245, 71, 297, 22, 317, 2, 5, 25, 5, 65, 29, 165, 5, 269, 81, 333, 25, 385, 109, 401, 2, 5, 5, 5, 61, 5, 153, 16, 5, 91, 377, 4, 449, 125, 61, 37, 509, 2
Offset: 1

Views

Author

Scott R. Shannon, Nov 22 2022

Keywords

Comments

The starting point can be any of the 4*n points around the square as changing the starting point simply rotates and/or reflects the resulting pattern formed by the path to one of the four orthogonal directions around the square; this does not change the number of regions formed by the path.
The number of times the path formed by the line touches and leaves the edges of the square is lcm(4*n,k)/k. For k >= n this is the number of points in the star-shaped pattern formed by the path.
The table starts with k = 2 as T(n,1) = 5 for all values of n. The maximum k is 2*n as T(n,2*n + m) = T(n,2*n - m).

Examples

			The table begins:
2;
5, 21, 2;
5,  5  4, 61,  2;
5, 29, 5, 73, 25, 105,  2;
5, 25, 5,  5, 31, 141, 11, 157,  2;
5,  5, 5, 85,  5, 153,  4,  25, 61, 229,  2;
5, 25, 5, 73, 33,   5, 15, 245, 71, 297, 22, 317,   2;
5, 25, 5, 65, 29, 165,  5, 269, 81, 333, 25, 385, 109, 401,  2;
5,  5, 5, 61,  5, 153, 16,   5, 91, 377,  4, 449, 125,  61, 37, 509,   2;
5, 25, 5,  5, 25, 137,  5, 285,  5, 385, 31, 501, 141,  25, 11, 613, 169, 629, 2;
.
.
See the attached file for more examples.

Links

Crossrefs

Cf. A358574 (vertices), A358627 (edges), A331452, A355798, A355838, A357058, A358407, A345459.

Formula

T(n,k) = A358627(n,k) - A358574(n,k) + 1 by Euler's formula.
T(n,2*n) = 2. The line cuts the square into two parts.
T(n,k) = 5 where n >= 2, k <= n, and k|(4*n). Four lines cut across the square's corners so four additional triangles are created.

A358574 Triangle read by rows: T(n,k) is the number of vertices formed when n points are placed along each edge of a square that divide the edges into n+1 equal parts and a line is continuously drawn from the current point to that k points, 2 <= k <= 2*n, counterclockwise around the square until the starting point is again reached.

Original entry on oeis.org

8, 12, 20, 12, 16, 16, 16, 64, 16, 20, 36, 20, 68, 36, 100, 20, 24, 36, 24, 24, 44, 144, 29, 144, 24, 28, 28, 28, 92, 28, 140, 28, 44, 76, 208, 28, 32, 44, 32, 84, 52, 32, 39, 240, 88, 292, 46, 296, 32, 36, 48, 36, 80, 52, 164, 36, 252, 100, 316, 52, 368, 124, 364, 36, 40, 40, 40, 80, 40, 164, 47, 40, 112, 364, 40, 436, 144, 88, 67, 472, 40
Offset: 1

Views

Author

Scott R. Shannon, Nov 23 2022

Keywords

Comments

See A358556 for further details.

Examples

			The table begins:
8;
12, 20, 12;
16, 16, 16, 64, 16;
20, 36, 20, 68, 36, 100, 20;
24, 36, 24, 24, 44, 144, 29, 144, 24;
28, 28, 28, 92, 28, 140, 28,  44,  76, 208, 28;
32, 44, 32, 84, 52,  32, 39, 240,  88, 292, 46, 296,  32;
36, 48, 36, 80, 52, 164, 36, 252, 100, 316, 52, 368, 124, 364, 36;
40, 40, 40, 80, 40, 164, 47,  40, 112, 364, 40, 436, 144,  88, 67, 472, 40;
.
.
See the attached file for more examples.

Links

Crossrefs

Cf. A358556 (regions), A358627 (edges), A331452, A355798, A355838, A357058, A358407, A345459.

Formula

T(n,k) = A358627(n,k) - A358556(n,k) + 1 by Euler's formula.
T(n,2*n) = 4*(n + 1). The line cuts the square into two parts so no new vertices are created.
T(n,k) = 4*(n + 1) where k <= n and k|(4*n). Four lines cut across the square's corners so no new vertices are created.

A358627 Triangle read by rows: T(n,k) is the number of edges formed when n points are placed along each edge of a square that divide the edges into n+1 equal parts and a line is continuously drawn from the current point to that k points, 2 <= k <= 2*n, counterclockwise around the square until the starting point is again reached.

Original entry on oeis.org

9, 16, 40, 13, 20, 20, 19, 124, 17, 24, 64, 24, 140, 60, 204, 21, 28, 60, 28, 28, 74, 284, 39, 300, 25, 32, 32, 32, 176, 32, 292, 31, 68, 136, 436, 29, 36, 68, 36, 156, 84, 36, 53, 484, 158, 588, 67, 612, 33, 40, 72, 40, 144, 80, 328, 40, 520, 180, 648, 76, 752, 232, 764, 37, 44, 44, 44, 140, 44, 316, 62, 44, 202, 740, 43, 884, 268, 148, 103, 980, 41
Offset: 1

Views

Author

Scott R. Shannon, Nov 24 2022

Keywords

Comments

See A358556 for further details and images of the squares.

Examples

			The table begins:
9;
16, 40, 13;
20, 20, 19, 124, 17;
24, 64, 24, 140, 60, 204, 21;
28, 60, 28,  28, 74, 284, 39, 300,  25;
32, 32, 32, 176, 32, 292, 31,  68, 136, 436, 29;
36, 68, 36, 156, 84,  36, 53, 484, 158, 588, 67, 612,  33;
40, 72, 40, 144, 80, 328, 40, 520, 180, 648, 76, 752, 232, 764,  37;
44, 44, 44, 140, 44, 316, 62,  44, 202, 740, 43, 884, 268, 148, 103, 980, 41;
.
.
See the attached file for more examples.

Links

Crossrefs

Cf. A358556 (regions), A358574 (vertices), A331452, A355798, A355838, A357058, A358407, A345459.

Formula

T(n,k) = A358574(n,k) + A358556(n,k) - 1 by Euler's formula.
T(n,2*n) = 4*(n + 1) + 1. The line cuts the square into two parts so one additional edge is created.
T(n,k) = 4*(n + 2) where n >= 2, k <= n, and k|(4*n). Four lines cut across the square's corners so four additional edges are created.
Showing 1-8 of 8 results.

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

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