A367121 -id:A367121 - cp's OEIS frontend

A334698 a(n) is the total number of points (both boundary and interior points) in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter), where the interior points are counted with multiplicity.

5, 58, 375, 1376, 3685, 8130, 15743, 27760, 45621, 70970, 105655, 151728, 211445, 287266, 381855, 498080, 639013, 807930, 1008311, 1243840, 1518405, 1836098, 2201215, 2618256, 3091925, 3627130, 4228983, 4902800, 5654101, 6488610, 7412255, 8431168, 9551685, 10780346, 12123895, 13589280, 15183653, 16914370, 18788991

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 18 2020

An equivalent definition: Place n-1 points in general position on each side of a square, and join every pair of the 4*n+4 boundary points by a chord; sequence gives number of vertices in the resulting planar graph. "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. - Scott R. Shannon and N. J. A. Sloane, Nov 05 2023
Equivalently, this is A334697(n) + 4*n.
This is an upper bound on A331449.

Colin Barker, Table of n, a(n) for n = 1..1000
Scott R. Shannon, Colored illustration for n = 2
Scott R. Shannon, Illustration for n=3 showing interior vertices color-coded according to multiplicity.
Scott R. Shannon, "General position" image for n = 1.
Scott R. Shannon, "General position" image for n = 2.
Scott R. Shannon, "General position" image for n = 3.
Scott R. Shannon, "General position" image for n = 4.
Scott R. Shannon, "General position" image for n = 5.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A255011, A331449, A334690-A334698.

For the "general position" version, see also A367121 (regions), A367122 (edges), and A367117.

A334698[n_]:=n(17n^3-30n^2+19n+4)/2;Array[A334698,50] (* or *)
LinearRecurrence[{5,-10,10,-5,1},{5,58,375,1376,3685},50] (* Paolo Xausa, Nov 14 2023 *)

Vec(x*(5 + 33*x + 135*x^2 + 31*x^3) / (1 - x)^5 + O(x^40)) \\ Colin Barker, May 31 2020

Theorem: a(n) = n*(17*n^3-30*n^2+19*n+4)/2.
From Colin Barker, May 27 2020: (Start)
G.f.: x*(5 + 33*x + 135*x^2 + 31*x^3) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
(End)
a(n+1) = A367122(n) - A367121(n) + 1 by Euler's formula.

Edited by N. J. A. Sloane, Nov 13 2023

A367276 Place n equally spaced points on each side of a square, and join each of these points by a chord to the 3*n new points on the other three sides: sequence gives number of vertices in the resulting planar graph.

Original entry on oeis.org

4, 9, 69, 345, 1337, 2885, 7445, 12833, 23365, 36589, 64669, 80133, 138313, 176885, 233765, 312013, 455273, 513277, 741965, 819589, 1046245, 1310761, 1692961, 1772097, 2315289, 2713997, 3165125, 3552753, 4538845, 4602985, 6015561, 6432681, 7421345, 8550485, 9439621, 10063993, 12635769

Offset: 0

Scott R. Shannon, Nov 11 2023

nonn

We start with the four corner points of the square, 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 3*n chords to the points that were added to the other three edges. There are 6*n^2 chords.

Scott R. Shannon, Image for n = 1.
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 = 10.

Cf. A367277 (interior vertices), A367278 (regions), A367279 (edges).
If the 4*n points are placed "in general position" instead of uniformly, we get sequences A334698, A367121, A367122.
If the 4*n points are placed uniformly and we also draw chords from the four corner points of the square to these 4*n points, we get A255011, A331448, A331449, A334690.

a(n) = A367279(n) - A367278(n) + 1 (Euler).

A367122 Place n points in general position on each side of a square, and join every pair of the 4*n+4 boundary points by a chord; sequence gives number of edges in the resulting planar graph.

Original entry on oeis.org

8, 124, 780, 2816, 7480, 16428, 31724, 55840, 91656, 142460, 211948, 304224, 423800, 575596, 764940, 997568, 1279624, 1617660, 2018636, 2489920, 3039288, 3674924, 4405420, 5239776, 6187400, 7258108, 8462124, 9810080, 11313016, 12982380, 14830028, 16868224, 19109640, 21567356

Offset: 0

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

nonn

"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.

See A334698 and A367121 for images of the square.

Cf. A334698 (vertices), A367121 (regions), A331448, A367119.

Conjecture: a(n) = 17*n^4 + 38*n^3 + 37*n^2 + 24*n + 8.

a(n) = A334698(n+1) + A367121(n) - 1 by Euler's formula.

A366253 Table read by antidiagonals: Place k points in general position on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives number of regions in the resulting planar graph.

Original entry on oeis.org

1, 13, 4, 82, 67, 11, 307, 406, 206, 24, 841, 1441, 1216, 489, 50, 1891, 3796, 4211, 2835, 995, 80, 3718, 8299, 10901, 9672, 5671, 1802, 154, 6637, 15982, 23536, 24780, 19139, 10196, 3052, 220, 11017, 28081, 44906, 53109, 48686, 34166, 17011, 4810, 375

Offset: 3

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

"In general position" implies that the internal lines (or chords) formed from the n*k edge points only have simple intersections; there is no interior points where three or more such chords meet. Note that for even-n n-gons, with n>=6, the chords from the n corner points do create non-simple intersections.

Note that although the number of regions with a given number of edges 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 created from the edge-point chords remain simple.

			The table begins:
1, 13, 82, 307, 841, 1891, 3718, 6637, 11017, 17281, 25906, 37423, 52417,...
4, 67, 406, 1441, 3796, 8299, 15982, 28081, 46036, 71491, 106294, 152497,...
11, 206, 1216, 4211, 10901, 23536, 44906, 78341, 127711, 197426, 292436,...
24, 489, 2835, 9672, 24780, 53109, 100779, 175080, 284472, 438585, 648219,...
50, 995, 5671, 19139, 48686, 103825, 196295, 340061, 551314, 848471, 1252175,...
80, 1802, 10196, 34166, 86480, 183770, 346532, 599126, 969776, 1490570,...
154, 3052, 17011, 56611, 142696, 302374, 569017, 982261, 1588006, 2438416,...
220, 4810, 26705, 88495, 222400, 470270, 883585, 1523455, 2460620, 3775450,...
375, 7305, 40096, 132243, 331431, 699535, 1312620, 2260941, 3648943, 5595261,...
444, 10509, 57810, 190263, 475980, 1003269, 1880634, 3236775, 5220588, 8001165,...
781, 14938, 81082, 265747, 663391, 1396396, 2615068, 4497637, 7250257,...
952, 20335, 110439, 361354, 900844, 1894347, 3544975, 6093514, 9818424,...
1456, 27391, 147421, 480931, 1197076, 2514781, 4702741, 8079421, 13013056,...
1696, 35716, 192552, 627484, 1560352, 3275556, 6122056, 10513372,...
.
.
.

Scott R. Shannon, Image for T(5,3).
Scott R. Shannon, Image for T(6,2).
Scott R. Shannon, Image for T(8,2).
Scott R. Shannon, Image for T(10,2).

Cf. A367118 (first row), A367121 (second row), A007678 (first column), A367183 (vertices), A367190 (edges).

T(n,k) = A367190(n,k) - A367183(n,k) + 1 by Euler's formula.
Conjectured:
T(3,k) = A367118(k) = (9/4)*k^4 + 3*k^3 + (15/4)*k^2 + 3*k + 1.
T(4,k) = A367121(k) = (17/2)*k^4 + 19*k^3 + (43/2)*k^2 + 14*k + 4.
T(5,k) = (45/2)*k^4 + 60*k^3 + 70*k^2 + (85/2)*k + 11.
T(6,k) = (195/4)*k^4 + (285/2)*k^3 + (687/4)*k^2 + 102*k + 24.
T(7,k) = (371/4)*k^4 + 287*k^3 + (1421/4)*k^2 + 210*k + 50.
T(8,k) = 161*k^4 + 518*k^3 + 655*k^2 + 388*k + 80.
T(9,k) = 261*k^4 + 864*k^3 + (2223/2)*k^2 + (1323/2)*k + 154.
T(10,k) = (1605/4)*k^4 + (2715/2)*k^3 + (7085/4)*k^2 + 1060*k + 220.

cp's OEIS Frontend

A334698 a(n) is the total number of points (both boundary and interior points) in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter), where the interior points are counted with multiplicity.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A367276 Place n equally spaced points on each side of a square, and join each of these points by a chord to the 3*n new points on the other three sides: sequence gives number of vertices in the resulting planar graph.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A367122 Place n points in general position on each side of a square, and join every pair of the 4*n+4 boundary points by a chord; sequence gives number of edges in the resulting planar graph.

Views

Author

Keywords

Comments

Crossrefs

Formula

A366253 Table read by antidiagonals: Place k points in general position on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives number of regions in the resulting planar graph.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula