A135708 Minimal total number of edges in a polyhex consisting of n hexagonal cells.

6, 11, 15, 19, 23, 27, 30, 34, 38, 41, 45, 48, 52, 55, 59, 62, 66, 69, 72, 76, 79, 83, 86, 89, 93, 96, 99, 103, 106, 109, 113, 116, 119, 123, 126, 129, 132, 136, 139, 142, 146, 149, 152, 155, 159, 162, 165, 168, 172, 175, 178, 181, 185, 188, 191, 194, 198, 201, 204, 207, 210

Offset: 1

Author

N. J. A. Sloane, based on an email from Sascha Kurz, Mar 05 2008

Keywords

• nonn

Comments

The extremal examples were described by Y. S. Kupitz in 1991.

References

• Y. S. Kupitz, "On the maximal number of appearances of the minimal distance among n points in the plane", in Intuitive geometry: Proceedings of the 3rd international conference held in Szeged, Hungary, 1991; Amsterdam: North-Holland: Colloq. Math. Soc. Janos Bolyai. 63, 217-244.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A135711.

Programs

Magma

[3*n+Ceiling(Sqrt(12*n-3)): n in [1..65]]; // Vincenzo Librandi, Oct 30 2016

Mathematica

Table[3*n + Ceiling[Sqrt[12*n - 3]], {n,1,25}] (* G. C. Greubel, Oct 29 2016 *)

PARI

a(n) = 3*n + ceil(sqrt(12*n-3)); \\ Michel Marcus, Oct 30 2016

Python

from math import isqrt
def A135708(n): return 3*n+1+isqrt(12*n-4) # Chai Wah Wu, Jul 28 2022

Formula

a(n) = 3*n + ceiling(sqrt(12*n - 3)). - H. Harborth

2*a(n) - A135711(n) = 6n. - Tanya Khovanova, Mar 07 2008