cp's OEIS Frontend

A121149 Minimal number of vertices in a planar connected n-polyhex.

%I A121149 #42 Feb 16 2025 08:33:02
%S A121149 1,6,10,13,16,19,22,24,27,30,32,35,37,40,42,45,47,50,52,54,57,59,62,
%T A121149 64,66,69,71,73,76,78,80,83,85,87,90,92,94,96,99,101,103,106,108,110,
%U A121149 112,115,117,119,121,124,126,128,130,133,135,137,139,142,144,146,148,150,153,155,157,159,162,164,166,168,170,173,175,177,179,181,184,186,188,190,192,195,197,199,201,203,206,208,210,212,214,216,219,221,223,225,227,230,232,234,236
%N A121149 Minimal number of vertices in a planar connected n-polyhex.
%C A121149 a(4) appears to be wrong: the polyhex labeled "bee" on Weisstein's article has 14 vertices. - _Joerg Arndt_, Oct 05 2016. However, "bee" has 16 vertices when the two "interior" vertices are counted, i.e., those where three hexagons meet.  - _Felix Fröhlich_, Oct 05 2016
%C A121149 a(n) is also the size of the smallest polyhex with n disjoint holes. - _Luca Petrone_, Feb 28 2017
%C A121149 Also numbers found at the end of n-th hexagonal arc of 'graphene' number spiral (numbers in the nodes of planar net 6^3, starting with 1). See the "Illustration for the first 76 terms" link. - _Yuriy Sibirmovsky_, Oct 04 2016
%C A121149 From _Ya-Ping Lu_, Feb 19 2022: (Start)
%C A121149 For each n-polyhex (n>=3), an n-gon can be constructed by connecting the centers of external neighboring hexagons in the n-polyhex. If the n-gon is convex (n is indicated by * in the figure below), a(n+1) = a(n) + 3; otherwise, a(n+1) = a(n) + 2. For example, for n=3, triangle 1-2-3-1 is convex and a(4) = a(3) + 3 = 16. For n=17, heptagon 6-8-9-11-13-15-17-6 is nonconvex and a(18) = a(17) + 2 = 52.
%C A121149 .
%C A121149           49--50--51--52*-53
%C A121149           / \ / \ / \ / \ / \
%C A121149         48*-28--29--30*-31--54
%C A121149         / \ / \ / \ / \ / \ / \
%C A121149       47--27*-13--14*-15--32--55
%C A121149       / \ / \ / \ / \ / \ / \ / \
%C A121149     46--26--12*--4*--5*-16*-33*-56*
%C A121149     / \ / \ / \ / \ / \ / \ / \ / \
%C A121149   45--25--11---3*--1---6--17--34--57
%C A121149     \ / \ / \ / \ / \ / \ / \ / \ /
%C A121149     44*-24*-10*--2---7*-18--35--58
%C A121149       \ / \ / \ / \ / \ / \ / \ /
%C A121149       43--23---9---8*-19*-36--59
%C A121149         \ / \ / \ / \ / \ / \ /
%C A121149         42--22--21*-20--37*-60
%C A121149           \ / \ / \ / \ / \ /
%C A121149           41--40*-39--38--61*
%C A121149 (End)
%H A121149 Moriah Elkin, Gregg Musiker, and Kayla Wright, <a href="https://arxiv.org/abs/2305.15531">Twists of Gr(3,n) Cluster Variables as Double and Triple Dimer Partition Functions</a>, arXiv:2305.15531 [math.CO], 2023. See p. 18.
%H A121149 Luca Petrone, <a href="/A121149/a121149.pdf">Illustration showing a(3) - a(43)</a>
%H A121149 Yuriy Sibirmovsky, <a href="/A121149/a121149.png">Illustration for the first 76 terms</a>
%H A121149 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polyhex.html">Polyhex</a>.
%Y A121149 Essentially the same as A182617: a(n) = A182617(n) + 1.
%K A121149 nonn,more
%O A121149 0,2
%A A121149 _Alexander Adamchuk_, Aug 12 2006
%E A121149 More terms from _Luca Petrone_, Mar 19 2017
%E A121149 a(0)=1 added by _N. J. A. Sloane_, Mar 23 2017