

A047932


a(n) = floor(3*nsqrt(12*n3)).


5



0, 1, 3, 5, 7, 9, 12, 14, 16, 19, 21, 24, 26, 29, 31, 34, 36, 39, 42, 44, 47, 49, 52, 55, 57, 60, 63, 65, 68, 71, 73, 76, 79, 81, 84, 87, 90, 92, 95, 98, 100, 103, 106, 109, 111, 114, 117, 120, 122, 125, 128, 131, 133, 136, 139, 142, 144, 147, 150, 153, 156, 158, 161
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OFFSET

1,3


COMMENTS

a(n) = cumulative sum of number of new pennypenny contacts when putting pennies on a table following a spiral pattern. This is the maximum possible number of contacts.
a(n) is also the maximum number of times the minimum distance can occur among n points in the plane [Harborth].


REFERENCES

Brass, Peter. "The maximum number of second smallest distances in finite planar sets." Discrete & Computational Geometry 7.1 (1992): 371379.
H. Harborth, Solution to problem 644A, Elemente der Mathematik (EMS Publishing House) 29, 1415


LINKS

Peter Kagey, Table of n, a(n) for n = 1..10000
K. Bezdek, M. A. Khan, Contact numbers for sphere packings, arXiv:1601.00145, Theorem 3.1.
R. W. GrosseKunstleve, Penny Spiral Sequence
MathOverflow, Maximal number of edges and triangular cells for n points in a triangular lattice, August 2011.


FORMULA

a(n) = floor(3*nsqrt(12*n3)).


MATHEMATICA

Table[Floor[3nSqrt[12n3]], {n, 70}] (* Harvey P. Dale, Dec 25 2014 *)


CROSSREFS

Partial sums of A047931.
A186705 is the maximum number of times the *same* distance can occur between n points in the plane, not necessarily the *minimum*.
Cf. A293956.
Sequence in context: A033036 A198082 A082767 * A139130 A219087 A186705
Adjacent sequences: A047929 A047930 A047931 * A047933 A047934 A047935


KEYWORD

nonn


AUTHOR

Ralf W. GrosseKunstleve


EXTENSIONS

Entry revised by N. J. A. Sloane, Nov 01 2017


STATUS

approved



