

A121149


Minimal number of vertices in a planar connected npolyhex.


5



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
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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 nth 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
For each npolyhex (n>=3), an ngon can be constructed by connecting the centers of external neighboring hexagons in the npolyhex. If the ngon 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 1231 is convex and a(4) = a(3) + 3 = 16. For n=17, heptagon 689111315176 is nonconvex and a(18) = a(17) + 2 = 52.
.
49505152*53
/ \ / \ / \ / \ / \
48*282930*3154
/ \ / \ / \ / \ / \ / \
4727*1314*153255
/ \ / \ / \ / \ / \ / \ / \
462612*4*5*16*33*56*
/ \ / \ / \ / \ / \ / \ / \ / \
4525113*16173457
\ / \ / \ / \ / \ / \ / \ / \ /
44*24*10*27*183558
\ / \ / \ / \ / \ / \ / \ /
432398*19*3659
\ / \ / \ / \ / \ / \ /
422221*2037*60
\ / \ / \ / \ / \ /
4140*393861*
(End)


LINKS

Eric Weisstein's World of Mathematics, Polyhex.


CROSSREFS



KEYWORD

nonn,more


AUTHOR



EXTENSIONS



STATUS

approved



