A047932 a(n) = floor(3*n-sqrt(12*n-3)).

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

Offset: 1

Author

Ralf W. Grosse-Kunstleve

Keywords

• nonn

Comments

a(n) = cumulative sum of number of new penny-penny 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].

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 [math.MG], 2016, Theorem 3.1.
• Peter Brass, The maximum number of second smallest distances in finite planar sets, Discrete & Computational Geometry 7.1 (1992): 371-379.
• R. W. Grosse-Kunstleve, Penny Spiral Sequence
• H. Harborth, Solution to problem 644A, Elemente der Mathematik (EMS Publishing House) 29, 14-15.
• MathOverflow, Maximal number of edges and triangular cells for n points in a triangular lattice, August 2011.

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.

Programs

Mathematica

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

Formula

a(n) = floor(3*n-sqrt(12*n-3)).

Extensions

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