A047932 -id:A047932 - cp's OEIS frontend

A047931 Number of new penny-penny contacts when putting pennies on a table following a spiral pattern.

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

Offset: 1

Ralf W. Grosse-Kunstleve

nonn

			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)

R. W. Grosse-Kunstleve, Penny Spiral Sequence

Cf. A047932.

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.

A186705 The Erdős unit distance problem: the maximum number of occurrences of the same distance among n points in the plane.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 12, 14, 18, 20, 23, 27, 30, 33, 37, 41, 43, 46, 50, 54, 57

Offset: 1

Michael Somos, Feb 25 2011

An upper bound is floor(k*n^(4/3)), A129011, if k is close enough to 1. Also a(27)=81 (Hamming 3,3 graph). - Ed Pegg Jr, Feb 02 2018

			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)

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

Boris Alexeev, Dustin G. Mixon, and Hans Parshall, The Erdős unit distance problem for small point sets, arXiv:2412.11914 [math.CO], 2024. See pp. 1, 12.
Thomas Bloom, Does every set of n distinct points in R^2 contain at most n^(1+O(1/log log n)) many pairs which are distince 1 apart?, Erdős Problems.
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.)
Paul 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
Terence Tao, Erdős problem database, see no. 90.
Eric Weisstein's World of Mathematics, Erdos Unit Distance Problem.
Eric Weisstein's World of Mathematics, Maximally Dense Unit-Distance Graph.

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

Cf. A385657 (number of nonisomorphic maximally dense unit-distance graphs).

Extended to a(21) using values from Version 2 of the Alexeev et al. arXiv manuscript. - N. J. A. Sloane, Jun 24 2025

A293956 Maximum over all sets of n points in the plane of the number of second-smallest distances between the points.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 01 2017

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).

Cf. A047932, A186705.

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.

Original entry on oeis.org

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

Offset: 1

Peter Kagey, May 03 2016

nonn

This is the hexagonal analog to A260643.

			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

Peter Kagey, Table of n, a(n) for n = 1..10000
Peter Kagey, Ruby program for computing sequence.

Cf. A047932, A260643.

A263135 The maximum number of penny-to-penny connections when n pennies are placed on the vertices of a hexagonal tiling.

Original entry on oeis.org

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

Offset: 0

Peter Kagey, Oct 10 2015

nonn

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.

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

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

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

Cf. A033581, A152743.

cp's OEIS Frontend

A047931 Number of new penny-penny contacts when putting pennies on a table following a spiral pattern.

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A186705 The Erdős unit distance problem: the maximum number of occurrences of the same distance among n points in the plane.

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A293956 Maximum over all sets of n points in the plane of the number of second-smallest distances between the points.

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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.

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A263135 The maximum number of penny-to-penny connections when n pennies are placed on the vertices of a hexagonal tiling.

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