OFFSET
0,2
COMMENTS
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
a(n) is also the size of the smallest polyhex with n disjoint holes. - Luca Petrone, Feb 28 2017
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
From Ya-Ping Lu, Feb 19 2022: (Start)
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.
.
49--50--51--52*-53
/ \ / \ / \ / \ / \
48*-28--29--30*-31--54
/ \ / \ / \ / \ / \ / \
47--27*-13--14*-15--32--55
/ \ / \ / \ / \ / \ / \ / \
46--26--12*--4*--5*-16*-33*-56*
/ \ / \ / \ / \ / \ / \ / \ / \
45--25--11---3*--1---6--17--34--57
\ / \ / \ / \ / \ / \ / \ / \ /
44*-24*-10*--2---7*-18--35--58
\ / \ / \ / \ / \ / \ / \ /
43--23---9---8*-19*-36--59
\ / \ / \ / \ / \ / \ /
42--22--21*-20--37*-60
\ / \ / \ / \ / \ /
41--40*-39--38--61*
(End)
LINKS
Moriah Elkin, Gregg Musiker, and Kayla Wright, Twists of Gr(3,n) Cluster Variables as Double and Triple Dimer Partition Functions, arXiv:2305.15531 [math.CO], 2023. See p. 18.
Luca Petrone, Illustration showing a(3) - a(43)
Yuriy Sibirmovsky, Illustration for the first 76 terms
Eric Weisstein's World of Mathematics, Polyhex.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Alexander Adamchuk, Aug 12 2006
EXTENSIONS
More terms from Luca Petrone, Mar 19 2017
a(0)=1 added by N. J. A. Sloane, Mar 23 2017
STATUS
approved