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 1-5 of 5 results.

A358408 Number of vertices formed in a square by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on each of the two adjacent edges of the square.

Original entry on oeis.org

4, 8, 32, 144, 468, 1160, 2512, 4836, 8468, 13760, 21784, 31168, 46596, 64760, 86912, 114656, 154400, 194116, 253160, 310408, 382712, 469296, 580688, 677656, 822928, 969880, 1141112, 1319984, 1566512, 1755032, 2080376, 2349188, 2686452, 3052184, 3450348, 3800756, 4387404, 4880560, 5443192
Offset: 1

Views

Author

Scott R. Shannon, Nov 14 2022

Keywords

Links

Crossrefs

Cf. A358407 (regions), A358409 (edges), A355799, A331449.

Formula

a(n) = A358409(n) - A358407(n) + 1 by Euler's formula.

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.

A358409 Number of edges formed in a square by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on each of the two adjacent edges of the square.

Original entry on oeis.org

4, 12, 68, 316, 1020, 2524, 5420, 10348, 18044, 29244, 45940, 66188, 97796, 135772, 182532, 240932, 321612, 405852, 525184, 646088, 796388, 974740, 1199244, 1407140, 1700944, 2004576, 2356296, 2729256, 3221296, 3630296, 4272656, 4835984, 5522768, 6269016, 7084056, 7835068, 8987192, 10005400
Offset: 1

Views

Author

Scott R. Shannon, Nov 14 2022

Keywords

Comments

See A358407 and A358408 for images of the square.

Crossrefs

Cf. A358407 (regions), A358408 (vertices), A355800, A331448.

Formula

a(n) = A358407(n) + A358408(n) - 1 by Euler's formula.
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.