A335351 -id:A335351 - cp's OEIS frontend

A335350 a(n) is the number of regions formed in a square by dividing each of its sides into n equal parts giving a total of 4n nodes and drawing straight line segments from node k to node (k+n+1) mod 4n, 0 <= k < 4*n.

4, 25, 37, 81, 109, 169, 205, 289, 341, 441, 485, 625, 701, 825, 913, 1089, 1189, 1369, 1461, 1661, 1805, 2025, 2141, 2389, 2549, 2809, 2929, 3249, 3405, 3721, 3901, 4205, 4421, 4753, 4913, 5329, 5549, 5913, 6105, 6561, 6781, 7225, 7453, 7885, 8189, 8649

Offset: 1

Lars Blomberg, Jun 03 2020

nonn

For n>1, a(n)-1 is divisible by 4.

Lars Blomberg, Table of n, a(n) for n = 1..500
Lars Blomberg, Illustration for n=3
Lars Blomberg, Illustration for n=4
Lars Blomberg, Illustration for n=5
Lars Blomberg, Illustration for n=10
Lars Blomberg, Illustration for n=16
Lars Blomberg, Illustration for n=35

Cf. A335351 (edges), A335352 (vertices), A335353 (n-gons), A335354 (edges in central polygon), A255011, A335057, A335192.

A335352 a(n) is the number of vertices formed in a square by dividing each of its sides into n equal parts giving a total of 4n nodes and drawing straight line segments from node k to node (k+n+1) mod 4n, 0 <= k < 4*n.

Original entry on oeis.org

5, 24, 28, 80, 100, 168, 188, 288, 324, 440, 452, 624, 676, 808, 872, 1088, 1156, 1368, 1412, 1644, 1764, 2024, 2084, 2380, 2500, 2808, 2848, 3248, 3340, 3720, 3836, 4188, 4356, 4744, 4800, 5328, 5476, 5896, 5984, 6560, 6684, 7224, 7348, 7852, 8100, 8648

Offset: 1

Lars Blomberg, Jun 04 2020

nonn

For n>1, a(n) is divisible by 4.

See A335350 for illustrations.

Lars Blomberg, Table of n, a(n) for n = 1..500

Cf. A335350 (regions), A335351 (edges), A335353 (n-gons), A335354 (edges in central polygon), A255011, A335057, A335192.

A335353 Irregular table read by rows: Take a square and divide each of its sides into n equal parts giving a total of 4n nodes, draw straight line segments from node k to node (k+n+1) mod 4n, 0 <= k < 4*n. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

Original entry on oeis.org

4, 16, 8, 0, 0, 0, 1, 32, 5, 32, 40, 8, 0, 0, 1, 64, 28, 16, 0, 0, 0, 0, 0, 0, 1, 80, 56, 24, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 96, 84, 24, 0, 0, 0, 0, 0, 0, 1, 128, 100, 40, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 144, 156, 32, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Lars Blomberg, Jun 04 2020

See A335350 for illustrations.

			Table begins:
4;
16, 8, 0, 0, 0, 1;
32, 5;
32, 40, 8, 0, 0, 1;
64, 28, 16, 0, 0, 0, 0, 0, 0, 1;
80, 56, 24, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
96, 84, 24, 0, 0, 0, 0, 0, 0, 1;
128, 100, 40, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
144, 156, 32, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
168, 188, 64, 16, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1;
200, 228, 40, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
248, 252, 88, 24, 8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

Lars Blomberg, Table of n, a(n) for n = 1..10000

Cf. A335350 (regions), A335351 (edges), A335352 (vertices), A335354 (edges in central polygon), A255011, A335057, A335192.

A335354 a(n) is the number of edges in the central polygon formed in a square by dividing each of its sides into n equal parts giving a total of 4n nodes and drawing straight line segments from node k to node (k+n+1) mod 4n, 0 <= k < 4*n.

Original entry on oeis.org

0, 8, 4, 8, 12, 16, 12, 16, 20, 16, 20, 24, 28, 24, 28, 32, 28, 32, 36, 32, 36, 40, 44, 40, 44, 48, 44, 48, 52, 56, 52, 56, 60, 56, 60, 64, 68, 64, 68, 72, 68, 72, 76, 72, 76, 80, 84, 80, 84, 88, 84, 88, 92, 96, 92, 96, 100, 96, 100, 104, 100, 104, 108, 112

Offset: 1

Lars Blomberg, Jun 04 2020

nonn

For n=1 there is no central polygon.
The number of edges of the central polygon tends to grow as n increases, whereas for n = 16..500 the polygon with next-to-most edges has 8 of them.
See A335350 for illustrations.

Lars Blomberg, Table of n, a(n) for n = 1..500

Cf. A335350 (regions), A335351 (edges), A335352 (vertices), A335353 (n-gons), A255011, A335057, A335192.

cp's OEIS Frontend

A335350 a(n) is the number of regions formed in a square by dividing each of its sides into n equal parts giving a total of 4n nodes and drawing straight line segments from node k to node (k+n+1) mod 4n, 0 <= k < 4*n.

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A335352 a(n) is the number of vertices formed in a square by dividing each of its sides into n equal parts giving a total of 4n nodes and drawing straight line segments from node k to node (k+n+1) mod 4n, 0 <= k < 4*n.

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A335353 Irregular table read by rows: Take a square and divide each of its sides into n equal parts giving a total of 4n nodes, draw straight line segments from node k to node (k+n+1) mod 4n, 0 <= k < 4*n. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

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A335354 a(n) is the number of edges in the central polygon formed in a square by dividing each of its sides into n equal parts giving a total of 4n nodes and drawing straight line segments from node k to node (k+n+1) mod 4n, 0 <= k < 4*n.

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cp's OEIS Frontend

A335350 a(n) is the number of regions formed in a square by dividing each of its sides into n equal parts giving a total of 4*n nodes and drawing straight line segments from node k to node (k+n+1) mod 4*n, 0 <= k < 4*n.

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A335352 a(n) is the number of vertices formed in a square by dividing each of its sides into n equal parts giving a total of 4*n nodes and drawing straight line segments from node k to node (k+n+1) mod 4*n, 0 <= k < 4*n.

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A335353 Irregular table read by rows: Take a square and divide each of its sides into n equal parts giving a total of 4*n nodes, draw straight line segments from node k to node (k+n+1) mod 4*n, 0 <= k < 4*n. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

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A335354 a(n) is the number of edges in the central polygon formed in a square by dividing each of its sides into n equal parts giving a total of 4*n nodes and drawing straight line segments from node k to node (k+n+1) mod 4*n, 0 <= k < 4*n.

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A335350 a(n) is the number of regions formed in a square by dividing each of its sides into n equal parts giving a total of 4n nodes and drawing straight line segments from node k to node (k+n+1) mod 4n, 0 <= k < 4*n.

A335352 a(n) is the number of vertices formed in a square by dividing each of its sides into n equal parts giving a total of 4n nodes and drawing straight line segments from node k to node (k+n+1) mod 4n, 0 <= k < 4*n.

A335353 Irregular table read by rows: Take a square and divide each of its sides into n equal parts giving a total of 4n nodes, draw straight line segments from node k to node (k+n+1) mod 4n, 0 <= k < 4*n. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

A335354 a(n) is the number of edges in the central polygon formed in a square by dividing each of its sides into n equal parts giving a total of 4n nodes and drawing straight line segments from node k to node (k+n+1) mod 4n, 0 <= k < 4*n.