1,1

The structure shows a hexagonal growth as in A182618.

a(n) is the number of vertices of the convex parts of the perimeter of the structure.

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

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

For n=1 there is 1 hexagon, so a(1)= 6 because there are 6 vertices that are connected to two edges.

For n=2 there are 2 connected hexagons, so a(2)= 8 because there are 8 vertices that are connected to two edges.

For n=3 there are 3 connected hexagons, so a(3)= 9 because there are 9 vertices that are connected to two edges.

If written as a triangle, begins:

6,

8,9,10,11,12,12,

13,14,14,15,15,16,16,17,17,18,18,18,

19,19,20,20,20,21,21,21,22,22,22,23,23,23,24,24,24,24

Row n has A008458(n-1) terms.

Cf. A182618, A257594.

Sequence in context: A096391 A381562 A190572 * A210447 A135558 A031951

Adjacent sequences: A182616 A182617 A182618 * A182620 A182621 A182622

nonn,more,tabf

Omar E. Pol, Dec 13 2010

approved