A047932 -id:A047932

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A047931

Number of new penny-penny contacts when putting pennies on a table following a spiral pattern.

+10
6

0, 1, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..91.

R. W. Grosse-Kunstleve, Penny Spiral Sequence

FORMULA

The n-th "chunk" consists of 2 3{n-2} 2 3{n-1} 2 3{n-1} 2 3{n-1} 2 3{n-1} 2 3{n}, where a{b} symbolizes b repetitions of a.

EXAMPLE

From Omar E. Pol, Nov 16 2016: (Start)

The sequence written as a spiral begins:

. 2 - 3 - 3 - 3 - 3 - 2

. / \

. 3 2 - 3 - 3 - 3 - 2 3

. / / \ \

. 3 3 2 - 3 - 3 - 2 3 3

. / / / \ \ \

. 3 3 3 2 - 3 - 2 3 3 3

. / / / / \ \ \ \

. 3 3 3 3 2 - 2 3 3 3 3

. / / / / / \ \ \ \ \

. 2 2 2 2 2 0 - 1 2 2 2 2

. \ \ \ \ \ / / / /

. 3 3 3 3 2 - 3 - 2 3 3 3

. \ \ \ \ / / /

. 3 3 3 2 - 3 - 3 - 2 3 3

. \ \ \ / /

. 3 3 2 - 3 - 3 - 3 - 2 3

. \ \ /

. 3 2 - 3 - 3 - 3 - 3 - 2

. \

. 2 - 3 - 3 - 3 - 3 - 3

(End)

CROSSREFS

Cf. A047932.

KEYWORD

nonn

AUTHOR

Ralf W. Grosse-Kunstleve

STATUS

approved

A186705

The maximum number of occurrences of the same distance among n points in the plane.

+10
4

0, 1, 3, 5, 7, 9, 12, 14, 18, 20, 23, 27, 30, 33

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

An upper bound is floor(k*n^(4/3)), A129011 if k is near enough to 1.

a(21)=57.

a(27)=81 (Hamming 3,3 graph). - Ed Pegg Jr, Feb 02 2018

REFERENCES

P. Brass, W. O. J. Moser, J. Pach, Research Problems in Discrete Geometry, Springer (2005), p. 183

LINKS

Table of n, a(n) for n=1..14.

Jean-Paul Delahaye, Les graphes-allumettes, (in French), Pour la Science no. 445, November 2014, pages 108-113. (On page 112, for n=6, drawing 2, one segment is missing.)

P. Erdős, On sets of distances of n points, American Mathematical Monthly 53, pp. 248-250 (1946).

Sascha Kurz, Plane point sets with many squares or isosceles right triangles, arXiv:2112.12716 [math.CO], 2021.

Ed Pegg Jr, Maximally Dense Unit Distance Graphs

EXAMPLE

a(4) = 5 because there is a unit distance graph with 4 vertices of an equilateral rhombus such that all but one of the six pairs of vertices are unit distance apart.

Comment from Allan C. Wechsler, Sep 17 2018: (Start)

Construction for a(9)=18: Take a convex, equilateral hexagon ABCDEF. Make the angles vary a bit, though, to avoid the hexagon being regular. Now, on each of the six sides, construct an equilateral triangle pointing into the hexagon. In general, the triangles will overlap here and there; this is OK because we aren't going to care about edges crossing each other. So we have triangles ABU, BCV, CDW, DEX, EFY, and FAZ: a total of twelve points with 18 unit distances among them.

Now adjust the hexagon to make some pairs of the internal points coincide. We want to make U=X, V=Y, and W=Z. The resulting linkage still has one degree of freedom, so we can arrange it so that none of the edges coincide (they can and must cross, though). The adjusted hexagon will only have two different angles: ABC = CDE = EFA, and BCD = DEF = FAB. The whole thing will have triangular (D_6) symmetry. It will have nine vertices (after merging three pairs from the original 12) but it will still have 18 unit edges. (End)

CROSSREFS

Cf. A047932, A051602, A063545, A129011, A186926, A293956.

KEYWORD

nonn,hard,more,nice

AUTHOR

Michael Somos, Feb 25 2011

STATUS

approved

A293956

Maximum over all sets of n points in the plane of the number of second-smallest distances between the points.

+10
3

0, 0, 2, 4, 6, 9, 11, 14, 18, 20

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..10.

Peter Brass, The maximum number of second smallest distances in finite planar sets, Discrete & Computational Geometry 7.1 (1992): 371-379.

Peter Brass, Annotated version of Fig. 1 from Brass (1992), illustrating small values of a(n).

CROSSREFS

Cf. A047932, A186705.

KEYWORD

nonn,more

AUTHOR

N. J. A. Sloane, Nov 01 2017

STATUS

approved

A272573

Start a spiral of numbers on a hexagonal tiling, with the initial hexagon as a(1) = 1. a(n) is the smallest positive integer not equal to or previously adjacent to its neighbors.

+10
2

1, 2, 3, 4, 5, 6, 7, 4, 6, 8, 5, 9, 8, 10, 2, 11, 3, 10, 11, 12, 13, 9, 12, 7, 13, 14, 1, 11, 13, 15, 9, 16, 14, 7, 16, 17, 15, 1, 16, 18, 7, 17, 19, 20, 1, 17, 18, 19, 9, 21, 3, 20, 10, 22, 4, 15, 21, 23, 5, 22, 23, 10, 21, 6, 22, 24, 25, 2, 14, 22, 25, 26, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

This is the hexagonal analog to A260643.

LINKS

Peter Kagey, Table of n, a(n) for n = 1..10000

Peter Kagey, Ruby program for computing sequence.

EXAMPLE

Illustration of a(1) through a(8) and a(13):

| | | | | | | | | 8 9 5

| | 3 | 4 3 | 4 3 | 4 3 | 4 3 | 4 3 | | 4 3 8

1 | 1 2 | 1 2 | 1 2 | 5 1 2 | 5 1 2 | 5 1 2 | 5 1 2 | ... | 5 1 2 6

| | | | | 6 | 6 7 | 6 7 4 | | 6 7 4

CROSSREFS

Cf. A047932, A260643.

KEYWORD

nonn

AUTHOR

Peter Kagey, May 03 2016

STATUS

approved

A263135

The maximum number of penny-to-penny connections when n pennies are placed on the vertices of a hexagonal tiling.

+10
1

0, 0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 30, 31, 32, 34, 35, 36, 38, 39, 41, 42, 43, 45, 46, 48, 49, 50, 52, 53, 55, 56, 57, 59, 60, 62, 63, 64, 66, 67, 69, 70, 72, 73, 74, 76, 77, 79, 80, 81, 83, 84, 86, 87, 89, 90

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

a(A033581(n)) = A152743(n).

1 <= a(n+1) - a(n) <=2 for all n > 0.

Lim_{n -> infinity} a(n)/n = 3/2.

Conjecture: a(2*n) - A047932(n) = A216256(n) for n > 0.

LINKS

Peter Kagey, Table of n, a(n) for n = 0..10000

EXAMPLE

. | | o o .

. | o o | o o o o .

. o o | o o o | o o o o .

. o o | o o | o o o o .

. | | o o o o .

. | | o o .

. | | .

. f(6) = 6 | f(10) = 11 | f(24) = 30 .

CROSSREFS

Cf. A047932 (triangular tiling), A123663 (square tiling).

Cf. A033581, A152743.

KEYWORD

nonn

AUTHOR

Peter Kagey, Oct 10 2015

STATUS

approved

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