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

1, 6, 10, 13, 16, 19, 22, 24, 27, 30, 32, 35, 37, 40, 42, 45, 47, 50, 52, 54, 57, 59, 62, 64, 66, 69, 71, 73, 76, 78, 80, 83, 85, 87, 90, 92, 94, 96, 99, 101, 103, 106, 108, 110, 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

Offset: 0

Author

Alexander Adamchuk, Aug 12 2006

Keywords

• nonn
• more

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

Essentially the same as A182617: a(n) = A182617(n) + 1.

Extensions

More terms from Luca Petrone, Mar 19 2017

a(0)=1 added by N. J. A. Sloane, Mar 23 2017