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%I #19 Feb 28 2017 09:27:21
%S 6,8,9,10,11,12,12,13,14,14,15,15,16,16,17,17,18,18,18,19,19,20,20,20,
%T 21,21,21,22,22,22,23,23,23,24,24,24,24
%N Number of vertices that are connected to two edges in a spiral without holes constructed with n hexagons.
%C The structure shows a hexagonal growth as in A182618.
%C a(n) is the number of vertices of the convex parts of the perimeter of the structure.
%C This sequence can be constructed geometrically in the following manner: Construct a gapless array of n equal circles with the rule of always choosing an arrangement with the maximum number of completely enclosed inner circles. Then, a(n) equals the number of circles required to create a kissing perimeter around the original array. Examples: a(1) = 6 because it takes 6 circles to create a kissing perimeter around 1 circle. a(7) = 12 because it takes 12 circles to create a kissing perimeter around 7 circles, which are arranged with 1 circle in center surrounded by 6 kissing circles. One could describe this as the "kissing numbers of kissing circles" sequence. - _Peter Woodward_, Apr 25 2015
%C a(n) is also the size of the smallest hexagonal polyomino that admits a hole of size n (Cf. A257594). - _Luca Petrone_, Feb 28 2017
%e For n=1 there is 1 hexagon, so a(1)= 6 because there are 6 vertices that are connected to two edges.
%e For n=2 there are 2 connected hexagons, so a(2)= 8 because there are 8 vertices that are connected to two edges.
%e For n=3 there are 3 connected hexagons, so a(3)= 9 because there are 9 vertices that are connected to two edges.
%e If written as a triangle, begins:
%e 6,
%e 8,9,10,11,12,12,
%e 13,14,14,15,15,16,16,17,17,18,18,18,
%e 19,19,20,20,20,21,21,21,22,22,22,23,23,23,24,24,24,24
%Y Cf. A182618. Row n has A008458(n-1) terms.
%K nonn,tabf
%O 1,1
%A _Omar E. Pol_, Dec 13 2010
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