A242088 -id:A242088 - cp's OEIS frontend

A080865 Order of symmetry groups of n points on 3-dimensional sphere with minimal distance between them maximized, also known as hostile neighbor or Tammes problem.

24, 12, 48, 6, 16, 12, 4, 10, 120, 8, 8

Offset: 4

Hugo Pfoertner, Feb 21 2003

If more than one best packing exists (this occurs for n = 15, 62, 76, 117, ...; see Buddenhagen, Kottwitz link) for a given n, the one with the largest symmetry group is chosen. A conjectured (except n=24) continuation of the sequence starting with n=15 would be: 3 16 4 2 2 12 1 1 1 24 3 2 4 1 1 6 5 6 3 2 1 4 2 24 1 3 1 10 1 2 1 2 1 24 2 12.

L. Fejes Toth, Lagerungen in der Ebene auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972.

James Buddenhagen and D. A. Kottwitz, Multiplicity and Symmetry Breaking in (Conjectured) Densest Packings of Congruent Circles on a Sphere.
D. A. Kottwitz, The Densest Packing of Equal Circles on a Sphere, Acta Cryst. (1991). A47, 158-165
O. R. Musin and A. S. Tarasov, The strong thirteen spheres problem, Discrete Comput. Geom., 48 (2012), 128-141, arXiv:1002.1439 [math.MG], 2010-2012.
O. R. Musin and A. S. Tarasov, The Tammes problem for N=14, Experimental Mathematics, 24 (2015), 460-468, arXiv:1410.2536 [math.MG].
Hugo Pfoertner, Arrangement of points on a sphere. Visualization of the best known solutions of the Tammes problem.
K. Schuette and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Annalen, 123 (1951), 96-124.
K. Schuette and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Annalen, 123 (1951), 96-124.
N. J. A. Sloane, Library of 3-d packings

A080866 gives the number of shortest edges which make up the rigid framework of the arrangement.
Cf. A081314, A084827, A242088, A242617, A217695, A268487.
Cf. A342559 (point numbers where records of packing density occur).

A242617 Decimal expansion of Kuijlaars-Saff constant, a constant related to Tammes' constants, Thomson's electron problem and Fekete points.

Original entry on oeis.org

5, 5, 3, 0, 5, 1, 2, 9, 3, 3, 5, 7, 5, 9, 5, 1, 8, 6, 7, 7, 9, 9, 5, 1, 0, 3, 7, 0, 8, 7, 1, 2, 4, 7, 7, 4, 5, 5, 0, 8, 0, 2, 8, 5, 7, 6, 0, 1, 9, 6, 6, 1, 7, 7, 6, 3, 3, 0, 4, 0, 7, 0, 9, 7, 0, 5, 9, 5, 3, 8, 7, 8, 8, 4, 0, 7, 7, 1, 2, 5, 4, 1, 6, 8, 7, 0, 5, 3, 7, 3, 2, 6, 3, 1, 6, 8, 2, 9, 1, 3

Offset: 0

Jean-François Alcover, May 19 2014

			-0.5530512933575951867799510370871247745508...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 8.8 p. 509.

A. B. J. Kuijlaars and E. B. Saff, Asymptotics for minimal discrete energy on the sphere
Eric Weisstein's World of Mathematics, Thomson Problem
Wikipedia, Thomson problem

Cf. A059750, A080865, A242088.

c = Sqrt[3]*Sqrt[Sqrt[3]/(8*Pi)]*Zeta[1/2]*(Zeta[1/2, 1/3] - Zeta[1/2, 2/3]); RealDigits[c, 10, 100] // First

sqrt(sqrt(27)/8/Pi)*zeta(1/2)*(zetahurwitz(1/2, 1/3) - zetahurwitz(1/2, 2/3)) \\ Charles R Greathouse IV, Jan 31 2018

sqrt(3)*sqrt(sqrt(3)/(8*Pi))*zeta(1/2)*(zeta(1/2, 1/3) - zeta(1/2, 2/3)).

A296515 Number of edges in a maximal planar graph with n vertices.

Original entry on oeis.org

0, 0, 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174

Offset: 0

Joseph Wheat, Jonathan Sondow, Feb 28 2018

{a(n)} is a monotonic increasing sequence because a maximal planar graph of order n can be generated on n + 1 nodes. Therefore a(n) <= a(n + 1).
Maximal planar graphs of order n > 5 are not unique.
|E(G_2n)| = (2n - 1) + 2*Sum_{k=0..(floor(log_2(n - 1)))} floor((n - 1)/2^k) where |E(G_2n)| is the size of a minimal planar graph G of order 2n.
Number of edges of a maximal 3-degenerate graph of order n (this class includes 3-trees).  The intersection of this class and maximal planar graphs is the Apollonian networks (planar 3-trees); neither class contains the other. - Allan Bickle, Nov 14 2021
a(n) is the number of blocks to dig (in a staircase fashion) to get out of a hole of depth n in Minecraft. - Max R Anderson, Oct 19 2023

Stefano Spezia, Table of n, a(n) for n = 0..10000
Allan Bickle, Structural results on maximal k-degenerate graphs, Discuss. Math. Graph Theory 32 4 (2012), 659-676.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Dawei He, Yan Wang, and Xingxing Yu, The Kelmans-Seymour conjecture I, arXiv:1511.05020 [math.CO], 2015.
D. R. Lick and A. T. White, k-degenerate graphs, Canad. J. Math. 22 (1970), 1082-1096.
Mathematics Stack Exchange, You dig a hole in Minecraft straight down n blocks, how many blocks must you dig to get out?
K. Stephenson, Circle Packing: A Mathematical Tale, Notices Amer. Math. Soc. 50 (2003).
Squid Tamar-Mattis, Planar Graphs and Wagner's and Kuratowski's Theorems, REU (2016).
Eric Weisstein's World of Mathematics, Kuratowski Reduction Theorem, Polyhedral Graph
David R. Wood, A Simple Proof of the Fary-Wagner Theorem, arXiv:cs/0505047 [math.CO], 2005.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008486, A008585, A242088.

CoefficientList[Series[(x^2 (1 + x + x^2))/(x - 1)^2, {x, 0, 60}], x] (* or *) LinearRecurrence[{2, -1}, {0, 0, 1, 3}, 60] (* Robert G. Wilson v, Mar 04 2018 *)

a(n) = floor(6/2^n) + 3n - 6 (see comments section of A008486).
G.f.: x^2 + 3*x^3/(x - 1)^2. - R. J. Mathar, Apr 14 2018
E.g.f.: 6 + x*(x + 6)/2 + 3*exp(x)*(x - 2). - Stefano Spezia, Feb 13 2023
a(n) = 3*(n - 2) for n >= 3. - Max R Anderson, Oct 19 2023

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A080865 Order of symmetry groups of n points on 3-dimensional sphere with minimal distance between them maximized, also known as hostile neighbor or Tammes problem.

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A242617 Decimal expansion of Kuijlaars-Saff constant, a constant related to Tammes' constants, Thomson's electron problem and Fekete points.

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A296515 Number of edges in a maximal planar graph with n vertices.

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