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 1-8 of 8 results.

A367015 Number of regions formed after n points have been placed in general position on each edge of the triangle in A365929.

Original entry on oeis.org

1, 4, 28, 136, 445, 1126, 2404, 4558, 7921, 12880, 19876, 29404, 42013, 58306, 78940, 104626, 136129, 174268, 219916, 274000, 337501, 411454, 496948, 595126, 707185, 834376, 978004, 1139428, 1320061, 1521370, 1744876, 1992154, 2264833, 2564596, 2893180, 3252376, 3644029, 4070038
Offset: 0

Views

Author

Scott R. Shannon, Nov 01 2023

Keywords

Comments

See A365929 for more information.

Links

  • Scott R. Shannon, Image for n = 2.
  • Scott R. Shannon, Image for n = 3. Note that although the number of k-gons will vary as the edge points change position the total number of regions will stay constant (at 136 for n = 3) as long as all internal vertices remain simple.

Crossrefs

Cf. A365929 (internal vertices), A366932 (edges), A366478 (vertices/3).

Formula

Conjecture: a(n) = (9*n^4 - 12*n^3 + 15*n^2 + 4)/4.
a(n) = A366932(n) - 3*A366478(n) + 1 by Euler's formula.

A366932 Number of edges formed after n points have been placed in general position on each edge of the triangle in A365929.

Original entry on oeis.org

3, 9, 51, 255, 855, 2193, 4719, 8991, 15675, 25545, 39483, 58479, 83631, 116145, 157335, 208623, 271539, 347721, 438915, 546975, 673863, 821649, 992511, 1188735, 1412715, 1666953, 1954059, 2276751, 2637855, 3040305, 3487143, 3981519, 4526691, 5126025, 5782995, 6501183, 7284279
Offset: 0

Views

Author

Scott R. Shannon, Nov 02 2023

Keywords

Comments

See A365929 for more information. See A365929 and A367015 for images of the triangle.

Crossrefs

Cf. A365929 (internal vertices), A367015 (regions), A366478.

Formula

Conjecture: a(n) = 3*(3/2*n^4 - 2*n^3 + 3/2*n^2 + n + 1).
a(n) = A367015(n) + 3*A366478(n) - 1 by Euler's formula.

A367117 Place n points in general position on each side of an equilateral triangle, and join every pair of the 3*n+3 boundary points by a chord; sequence gives number of vertices in the resulting planar graph.

Original entry on oeis.org

3, 12, 72, 282, 795, 1818, 3612, 6492, 10827, 17040, 25608, 37062, 51987, 71022, 94860, 124248, 159987, 202932, 253992, 314130, 384363, 465762, 559452, 666612, 788475, 926328, 1081512, 1255422, 1449507, 1665270, 1904268, 2168112, 2458467, 2777052, 3125640, 3506058, 3920187, 4369962
Offset: 0

Views

Author

Scott R. Shannon and N. J. A. Sloane, Nov 05 2023

Keywords

Comments

"In general position" implies that the internal lines (or chords) only have simple intersections. There is no interior point where three or more chords meet.
Note that although the number of k-gons in the graph will vary as the edge points change position, the total number of regions will stay constant as long as all internal vertices remain simple.

Links

Crossrefs

Cf. A367118 (regions), A367119 (edges).
See also A091908, A092098, A331782, A365929.
If the boundary points are equally spaced, we get A274585, A092866, A274586, A092867. - N. J. A. Sloane, Nov 09 2023

Programs

Formula

Theorem: a(n) = (3/4)*(n+1)*(3*n^3+n^2+4).
a(n) = A367119(n) - A367118(n) + 1 by Euler's formula.

A366478 a(n) = (3*n^4 - 4*n^3 + n^2 + 4*n + 4)/4.

Original entry on oeis.org

1, 2, 8, 40, 137, 356, 772, 1478, 2585, 4222, 6536, 9692, 13873, 19280, 26132, 34666, 45137, 57818, 73000, 90992, 112121, 136732, 165188, 197870, 235177, 277526, 325352, 379108, 439265, 506312, 580756, 663122, 753953, 853810, 963272, 1082936, 1213417, 1355348, 1509380
Offset: 0

Views

Author

N. J. A. Sloane, Oct 29 2023

Keywords

Comments

This is one-third of the total number of points in the configuration obtained at the n-th stage of A365929.

Links

Crossrefs

Cf. A365929.

Programs

Formula

G.f.: -(2*x^4+10*x^3+8*x^2-3*x+1)/(x-1)^5. - Alois P. Heinz, Nov 02 2023

A366483 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of vertices in the resulting planar graph.

Original entry on oeis.org

3, 6, 22, 108, 300, 919, 1626, 3558, 5824, 9843, 14352, 23845, 30951, 47196, 62773, 82488, 104544, 144784, 173694, 230008, 276388, 336927, 403452, 509218, 582417, 702228, 824956, 969387, 1098312, 1321978, 1463580, 1724190, 1952509, 2221497, 2505169, 2846908, 3103788, 3556143, 3978763, 4444003
Offset: 0

Views

Author

Scott R. Shannon and N. J. A. Sloane, Nov 09 2023

Keywords

Comments

We start with the three corner points of the triangle, and add n further points along each edge. Including the corner points, we end up with n+2 points along each edge, and the edge is divided into n+1 line segments.
Each of the n points added to an edge is joined by 2*n chords to the points that were added to the other two edges. There are 3*n^2 chords.

Links

Crossrefs

Cf. A366484 (interior vertices), A366485 (edges), A366486 (regions).
If the 3*n points are placed "in general position" instead of uniformly, we get sequences A366478, A365929, A366932, A367015.
If the 3*n points are placed uniformly and we also draw chords from the three corner points of the triangle to these 3*n points, we get A274585, A092866, A274586, A092867.

Formula

a(n) = A366485(n) - A366486(n) + 1 (Euler).

A366484 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of interior vertices in the resulting planar graph.

Original entry on oeis.org

0, 0, 13, 96, 285, 901, 1605, 3534, 5797, 9813, 14319, 23809, 30912, 47154, 62728, 82440, 104493, 144730, 173637, 229948, 276325, 336861, 403383, 509146, 582342, 702150, 824875, 969303, 1098225, 1321888, 1463487, 1724094, 1952410, 2221395, 2505064, 2846800, 3103677, 3556029, 3978646, 4443883
Offset: 0

Views

Author

Scott R. Shannon and N. J. A. Sloane, Nov 09 2023

Keywords

Comments

See A366483 for further information.

Links

Crossrefs

Cf. A366483 (vertices), A366485 (edges), A366486 (regions).
If the 3*n points are placed "in general position" instead of uniformly, we get sequences A366478, A365929, A366932, A367015.
If the 3*n points are placed uniformly and we also draw chords from the three corner points of the triangle to these 3*n points, we get A274585, A092866, A274586, A092867.

Formula

a(n) = A366485(n) - A366486(n) - 3*n - 2 (Euler).

A366486 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of regions in the resulting planar graph.

Original entry on oeis.org

1, 4, 27, 130, 385, 1044, 2005, 4060, 6831, 11272, 16819, 26436, 35737, 52147, 69984, 92080, 117952, 157770, 193465, 249219, 302670, 368506, 443026, 546462, 635125, 757978, 890133, 1041775, 1191442, 1407324, 1581058, 1837417, 2085096, 2365657, 2670429, 3018822, 3328351, 3771595, 4213602
Offset: 0

Views

Author

Scott R. Shannon and N. J. A. Sloane, Nov 09 2023

Keywords

Comments

See A366483 for further information.

Links

Crossrefs

Cf. A366483 (vertices), A366484 (interior vertices), A366485 (edges).
If the 3*n points are placed "in general position" instead of uniformly, we get sequences A366478, A365929, A366932, A367015.
If the 3*n points are placed uniformly and we also draw chords from the three corner points of the triangle to these 3*n points, we get A274585, A092866, A274586, A092867.

Formula

a(n) = A366485(n) - A366483(n) + 1 (Euler).

A366485 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of edges in the resulting planar graph.

Original entry on oeis.org

3, 9, 48, 237, 684, 1962, 3630, 7617, 12654, 21114, 31170, 50280, 66687, 99342, 132756, 174567, 222495, 302553, 367158, 479226, 579057, 705432, 846477, 1055679, 1217541, 1460205, 1715088, 2011161, 2289753, 2729301, 3044637, 3561606, 4037604, 4587153, 5175597, 5865729, 6432138, 7327737
Offset: 0

Views

Author

Scott R. Shannon and N. J. A. Sloane, Nov 09 2023

Keywords

Comments

See A366483 for further information. See A366483 and A366486 for images of the triangle.

Crossrefs

Cf. A366483 (vertices), A366484 (interior vertices), A366486 (regions).
If the 3*n points are placed "in general position" instead of uniformly, we get sequences A366478, A365929, A366932, A367015.
If the 3*n points are placed uniformly and we also draw chords from the three corner points of the triangle to these 3*n points, we get A274585, A092866, A274586, A092867.

Formula

a(n) = A366483(n) + A366486(n) - 1 (Euler).
Showing 1-8 of 8 results.

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

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