A368755 -id:A368755 - cp's OEIS frontend

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

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.

A368758 Irregular table read by rows: T(n,k) is the number of k-sided regions, k>=3, in the hyperoctahedral (or cocktail party) graph of order n.

Original entry on oeis.org

0, 2, 14, 2, 2, 42, 22, 100, 72, 12, 2, 234, 142, 4, 418, 320, 90, 10, 734, 610, 116, 44, 1248, 878, 82, 20, 14, 1968, 1454, 534, 98, 12, 16, 2696, 2662, 744, 74, 56, 34, 4040, 3434, 770, 74, 0, 2, 5806, 4722, 1932, 430, 94, 26, 7706, 7102, 2048, 894, 92, 24, 10868, 7492, 1448, 406, 4

Offset: 1

Scott R. Shannon, Jan 04 2024

See A368755 and A368756 for further images of the graphs.

			The table begins:
0;
2;
14, 2, 2;
42, 22;
100, 72, 12, 2;
234, 142, 4;
418, 320, 90, 10;
734, 610, 116, 44;
1248, 878, 82, 20, 14;
1968, 1454, 534, 98, 12, 16;
2696, 2662, 744, 74, 56, 34;
4040, 3434, 770, 74, 0, 2;
5806, 4722, 1932, 430, 94, 26;
7706, 7102, 2048, 894, 92, 24;
10868, 7492, 1448, 406, 4;
13438, 12122, 4682, 1356, 206, 4;
17438, 15950, 5420, 2194, 296, 84, 6, 2;
22990, 17734, 7166, 1976, 182, 52;
27284, 25902, 9672, 2718, 772, 182;
34160, 31164, 12650, 3710, 648, 188, 0, 0, 8, 32;
.
.

Scott R. Shannon, Image for T(8,k).
Eric Weisstein's World of Mathematics, Cocktail Party Graph.

Cf. A368755 (regions), A368756 (vertices), A368757 (edges), A129348, A193130, A282010.

Sum of row n = A368755(n).

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.

A368813 Number of regions 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, 1, 13, 57, 171, 361, 813, 1489, 2215, 4081, 6249, 8329, 13027, 17977, 20221, 32033, 41583, 50545, 66881, 83161, 93871, 124521, 150237, 173233, 213351, 251473, 285445, 342889, 397011, 432121, 524149, 598081, 663103, 769217, 867441, 960121, 1091723, 1218889, 1333489, 1506321, 1667799

Offset: 1

Scott R. Shannon, Jan 06 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 = 10.
Scott R. Shannon, Image for n = 15.

Cf. A368814 (vertices), A368815 (edges), A368816 (k-gons), A368755, A007678.

a(n) = A368815(n) - A368814(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).

A369175 Number of regions 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, 12, 36, 86, 180, 330, 570, 918, 1408, 2058, 2946, 4054, 5502, 7278, 9430, 12006, 15174, 18846, 23268, 28338, 34172, 40806, 48546, 57174, 67022, 78006, 90324, 103910, 119276, 135978, 154722, 175226, 197686, 222098, 248790, 277462, 309050, 343086, 379858, 419182, 462106, 507678

Offset: 1

Scott R. Shannon, Jan 15 2024

nonn

Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8. Note that 5-gons are now present.
Scott R. Shannon, Image for n = 15.

Cf. A369176 (vertices), A369177 (edges), A369178 (k-gons), A306302, A306302, A368755, A290131.

a(n) = A369177(n) - A369176(n) + 1 by Euler's formula.

cp's OEIS Frontend

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

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A368758 Irregular table read by rows: T(n,k) is the number of k-sided regions, k>=3, 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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A368813 Number of regions 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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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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Formula

A369175 Number of regions 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.

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Author

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Crossrefs

Formula