A126026 Conjectured upper bound on area of the convex hull of any edge-to-edge connected system of regular unit hexagons (n-polyhexes).

0, 1, 2, 4, 5, 8, 10, 13, 17, 20, 24, 28, 33, 38, 43, 49, 55, 61, 68, 75, 82, 90, 97, 106, 114, 123, 133, 142, 152, 162, 173, 184, 195, 207, 219, 231, 244, 257, 270, 284, 297, 312, 326, 341, 357, 372, 388, 404, 421, 438, 455, 473, 491, 509, 528, 547, 566

Offset: 0

Author

Jonathan Vos Post, Feb 27 2007

Keywords

• easy
• nonn

Comments

Kurz proved the polyomino equivalent of this conjecture as A122133 and abstracts: "In this article we prove a conjecture of Bezdek, Brass and Harborth concerning the maximum volume of the convex hull of any facet-to-facet connected system of n unit hypercubes in the d-dimensional Euclidean space. For d=2 we enumerate the extremal polyominoes and determine the set of possible areas of the convex hull for each n."

Examples

			a(10) = 24 because floor((10^2 + 14*10/3 + 1)/6) = floor(24.6111111) = 24.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Sascha Kurz, Convex hulls of polyominoes, arXiv:math/0702786 [math.CO], Feb 26 2007. See conjecture 2, p. 12.
• Eric Weisstein's World of Mathematics, Polyhex.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-2,1).

Crossrefs

Cf. A000228, A036359, A002216, A005963, A001998, A018190, A001207, A057973, A122133.

Programs

Mathematica

Table[Floor[(n^2+14n/3+1)/6],{n,0,80}] (* Harvey P. Dale, Apr 11 2012 *)

PARI

concat(0, Vec(x*(1 +x^2)*(1 -x^3 +2*x^4 -x^6 +x^7 +x^11 -x^13 +x^14 +x^15 -x^16) / ((1 -x)^3*(1 +x)*(1 -x +x^2)*(1 +x +x^2)*(1 -x^3 +x^6)*(1 +x^3 +x^6)) + O(x^50))) \\ Colin Barker, Oct 13 2016

PARI

a(n) = (n^2 + 14*n/3 + 1)\6 \\ Charles R Greathouse IV, Oct 13 2016

Formula

a(n) = floor((n^2 + 14*n/3 + 1)/6).

G.f.: x*(1 +x^2)*(1 -x^3 +2*x^4 -x^6 +x^7 +x^11 -x^13 +x^14 +x^15 -x^16) / ((1 -x)^3*(1 +x)*(1 -x +x^2)*(1 +x +x^2)*(1 -x^3 +x^6)*(1 +x^3 +x^6)). - Colin Barker, Oct 13 2016

Extensions

More terms from Harvey P. Dale, Apr 11 2012

Offset changed to 0 by Colin Barker, Oct 13 2016