A368758 -id:A368758 - cp's OEIS frontend

A368755 Number of regions in the hyperoctahedral (or cocktail party) graph of order n.

0, 2, 18, 64, 186, 380, 838, 1504, 2242, 4082, 6266, 8320, 13010, 17866, 20218, 31808, 41390, 50100, 66530, 82560, 93446, 123642, 149398, 171920, 212166, 249810, 283678, 340704, 394882, 428892, 521406, 594560, 659382, 764866, 863154, 954192, 1086490, 1212506, 1326654, 1498720, 1660278

Offset: 1

Scott R. Shannon, Jan 04 2024

nonn

Scott R. Shannon, Image for n = 2.
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.
Scott R. Shannon, Image for n = 9.
Scott R. Shannon, Image for n = 10.
Eric Weisstein's World of Mathematics, Cocktail Party Graph.

Cf. A368756 (vertices), A368757 (edges), A368758 (k-gons), A129348, A193130, A282010.

a(n) = A368757(n) - A368756(n) + 1 by Euler's formula.

A368756 Number of vertices in the hyperoctahedral (or cocktail party) graph of order n.

Original entry on oeis.org

2, 5, 17, 49, 151, 273, 693, 1249, 1711, 3525, 5529, 6777, 11711, 16133, 15937, 29121, 38227, 44561, 61985, 77041, 81423, 116165, 140997, 157649, 201211, 237125, 263449, 324689, 377359, 392185, 499789, 570241, 621255, 735493, 831537, 909097, 1048887, 1171013, 1265501, 1450081, 1608523

Offset: 1

Scott R. Shannon, Jan 04 2024

nonn

Scott R. Shannon, Image for n = 2.
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.
Scott R. Shannon, Image for n = 9.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 15. Note this 30-gon still contains vertices with 7 chords crossing, so this maximum possible value is the same as the regular n-gon with all diagonals drawn; see A007569.
Eric Weisstein's World of Mathematics, Cocktail Party Graph.

Cf. A368755 (regions), A368757 (edges), A368758 (k-gons), A007569, A129348, A193130, A282010.

a(n) = A368757(n) - A368755(n) + 1 by Euler's formula.

A368757 Number of edges in the hyperoctahedral (or cocktail party) graph of order n.

Original entry on oeis.org

0, 6, 34, 112, 336, 652, 1530, 2752, 3952, 7606, 11794, 15096, 24720, 33998, 36154, 60928, 79616, 94660, 128514, 159600, 174868, 239806, 290394, 329568, 413376, 486934, 547126, 665392, 772240, 821076, 1021194, 1164800, 1280636, 1500358, 1694690, 1863288, 2135376, 2383518, 2592154

Offset: 1

Scott R. Shannon, Jan 04 2024

nonn

See A368755 and A368756 for images of the graphs.

Eric Weisstein's World of Mathematics, Cocktail Party Graph.

Cf. A368755 (regions), A368756 (vertices), A368758 (k-gons), A129348, A193130, A282010.

a(n) = A368755(n) + A368756(n) - 1 by Euler's formula.

A368816 Irregular table read by rows: T(n,k) is the number of k-gons, k>=3, in a regular 2n-gon when all vertices are connect by straight lines except for the n lines joining diametrically opposite vertices.

Original entry on oeis.org

0, 0, 1, 12, 0, 0, 1, 40, 8, 8, 0, 0, 1, 100, 40, 20, 10, 0, 0, 0, 1, 204, 132, 12, 12, 0, 0, 0, 0, 0, 1, 378, 280, 126, 28, 0, 0, 0, 0, 0, 0, 0, 1, 688, 528, 176, 96, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1188, 864, 72, 54, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Scott R. Shannon, Jan 06 2024

See A368813 and A368814 for other images of the 2n-gons.

			The table begins:
0;
0, 1;
12, 0, 0, 1;
40, 8, 8, 0, 0, 1;
100, 40, 20, 10, 0, 0, 0, 1;
204, 132, 12, 12, 0, 0, 0, 0, 0, 1;
378, 280, 126, 28, 0, 0, 0, 0, 0, 0, 0, 1;
688, 528, 176, 96, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
1188, 864, 72, 54, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
1840, 1360, 620, 220, 20, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
2508, 2552, 880, 198, 44, 66, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
3648, 3576, 864, 216, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
.
.

Scott R. Shannon, Image for T(7,k).
Scott R. Shannon, Table for n = 1..100.

Cf. A368813 (regions), A368814 (vertices), A368815 (edges), A368755, A368758.

Sum of row n = A368813(n).

A369178 Irregular table read by rows: T(n,k) is the number of k-sided regions, k>=3, in a graph of n adjacent rectangles in a row with all possible diagonals drawn, as in A306302, but without the rectangles' edges which are perpendicular to the row.

Original entry on oeis.org

2, 8, 4, 22, 14, 52, 34, 98, 82, 184, 146, 302, 268, 484, 426, 8, 710, 694, 4, 1064, 986, 8, 1498, 1436, 12, 2056, 1986, 12, 2710, 2780, 12, 3624, 3630, 24, 4682, 4728, 20, 6012, 5970, 24, 7518, 7628, 28, 9408, 9406, 32, 11526, 11702, 40, 14028, 14246, 64, 16782, 17330, 60

Offset: 1

Scott R. Shannon, Jan 15 2024

Unlike the graph in A306302, or the complete bipartite graph of order n, for n>=8 the graph contains regions with 5 edges. It is likely 5 is the maximum number of edges in any region for all n.

			The table begins:
2;
8, 4;
22, 14;
52, 34;
98, 82;
184, 146;
302, 268;
484, 426, 8;
710, 694, 4;
1064, 986, 8;
1498, 1436, 12;
2056, 1986, 12;
2710, 2780, 12;
3624, 3630, 24;
4682, 4728, 20;
6012, 5970, 24;
7518, 7628, 28;
9408, 9406, 32;
11526, 11702, 40;
14028, 14246, 64;
16782, 17330, 60;
20220, 20518, 68;
23998, 24468, 80;
28304, 28786, 84;
.
.

Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8.

Cf. A369175 (regions), A369176 (vertices), A369177 (edges), A306302, A324042, A324043, A368758.

Sum of row(n) = A369175(n).

cp's OEIS Frontend

A368755 Number of regions in the hyperoctahedral (or cocktail party) graph of order n.

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A368756 Number of vertices in the hyperoctahedral (or cocktail party) graph of order n.

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A368757 Number of edges in the hyperoctahedral (or cocktail party) graph of order n.

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A368816 Irregular table read by rows: T(n,k) is the number of k-gons, k>=3, in a regular 2n-gon when all vertices are connect by straight lines except for the n lines joining diametrically opposite vertices.

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A369178 Irregular table read by rows: T(n,k) is the number of k-sided regions, k>=3, in a graph of n adjacent rectangles in a row with all possible diagonals drawn, as in A306302, but without the rectangles' edges which are perpendicular to the row.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula