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.

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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.

A355902 Start with a 2 X n array of squares, join every vertex on top edge to every vertex on bottom edge; a(n) = one-half the number of cells.

Original entry on oeis.org

0, 3, 10, 26, 56, 112, 196, 331, 522, 790, 1138, 1615, 2204, 2975, 3910, 5041, 6388, 8047, 9958, 12262, 14894, 17920, 21346, 25347, 29796, 34875, 40522, 46854, 53826, 61716, 70274, 79883, 90380, 101875, 114346, 127981, 142612, 158737, 176086, 194827, 214852, 236717, 259906, 285124, 311970, 340588, 370990, 403819, 438440, 475556
Offset: 0

Views

Author

Scott R. Shannon and N. J. A. Sloane, Sep 05 2022

Keywords

Comments

Note that this figure can be obtained by drawing an "equatorial" line through the middle of the strip of n adjacent rectangles in A306302. This cuts each of the 2n "equatorial" cells in A306302 in two. It follows that 2*a(n) = A306302(n) + 2*n, i.e. that a(n) = A306302(n)/2 + n. Note that there is an explicit formula for A306302(n) in terms of n. - Scott R. Shannon, Sep 06 2022.
This means the present sequence is one more member of the large class of sequences which are essentially the same as A115004 (see Cross-References). - N. J. A. Sloane, Sep 06 2022

Links

Crossrefs

Cf. A356790, A306302, A355798, A290131, A331452.
The following nine sequences are all essentially the same. The simplest is A115004(n), which we denote by z(n). Then A088658(n) = 4*z(n-1); A114043(n) = 2*z(n-1)+2*n^2-2*n+1; A114146(n) = 2*A114043(n); A115005(n) = z(n-1)+n*(n-1); A141255(n) = 2*z(n-1)+2*n*(n-1); A290131(n) = z(n-1)+(n-1)^2; A306302(n) = z(n)+n^2+2*n; A355902(n) = n + A306302(n)/2. - N. J. A. Sloane, Sep 06 2022

Formula

a(n) = A356790(2,n+2)/2 - 2.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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