cp's OEIS Frontend

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

%I A126026 #23 Feb 16 2025 08:33:04
%S A126026 0,1,2,4,5,8,10,13,17,20,24,28,33,38,43,49,55,61,68,75,82,90,97,106,
%T A126026 114,123,133,142,152,162,173,184,195,207,219,231,244,257,270,284,297,
%U A126026 312,326,341,357,372,388,404,421,438,455,473,491,509,528,547,566
%N A126026 Conjectured upper bound on area of the convex hull of any edge-to-edge connected system of regular unit hexagons (n-polyhexes).
%C A126026 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."
%H A126026 Colin Barker, <a href="/A126026/b126026.txt">Table of n, a(n) for n = 0..1000</a>
%H A126026 Sascha Kurz, <a href="http://arXiv.org/abs/math.CO/0702786">Convex hulls of polyominoes</a>, arXiv:math/0702786 [math.CO], Feb 26 2007. See conjecture 2, p. 12.
%H A126026 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polyhex.html">Polyhex.</a>
%H A126026 <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-2,1).
%F A126026 a(n) = floor((n^2 + 14*n/3 + 1)/6).
%F A126026 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
%e A126026 a(10) = 24 because floor((10^2 + 14*10/3 + 1)/6) = floor(24.6111111) = 24.
%t A126026 Table[Floor[(n^2+14n/3+1)/6],{n,0,80}] (* _Harvey P. Dale_, Apr 11 2012 *)
%o A126026 (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
%o A126026 (PARI) a(n) = (n^2 + 14*n/3 + 1)\6 \\ _Charles R Greathouse IV_, Oct 13 2016
%Y A126026 Cf. A000228, A036359, A002216, A005963, A001998, A018190, A001207, A057973, A122133.
%K A126026 easy,nonn
%O A126026 0,3
%A A126026 _Jonathan Vos Post_, Feb 27 2007
%E A126026 More terms from _Harvey P. Dale_, Apr 11 2012
%E A126026 Offset changed to 0 by _Colin Barker_, Oct 13 2016