

A247397


Numbers n such that when n unitdiameter circles are arranged nonoverlapping in the plane, and those circles are then enclosed in a rectangle, the area of the rectangle must be at least n.


0



1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13
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OFFSET

1,2


COMMENTS

For any number that does not appear on this list, there exists an arrangement of that number of unitdiameter circles that can be enclosed in a rectangle with area of less than 1 square unit per circle.
Any number of unitdiameter circles greater than or equal to 14 can be arranged in two rows, where the upper row is offset by 1/2 horizontally and (sqrt(3/4)1) vertically, thereby reducing the minimum size of the enclosing rectangle to less than n square units. However, this isn't necessarily the overall minimum.
In addition, 11 unitdiameter circles placed in 3 rows can be enclosed in an area less than 11 square units.


LINKS

Table of n, a(n) for n=1..12.
Eckard Specht, Densest known packings of a given number of circles in a rectangle


EXAMPLE

11 unitdiameter circles can be placed in a hexagonal array, with rows of 4, 3 and 4 circles in respective rows, which can be enclosed in a rectangle 4 units wide and (1+sqrt(3)) high, giving an area of 10.93, less than 11 square units. Any fewer circles than this, and also 12 or 13 circles, cannot be enclosed in a rectangle smaller than n square units in area.


CROSSREFS

Sequence in context: A276443 A239139 A228723 * A194845 A194056 A020753
Adjacent sequences: A247394 A247395 A247396 * A247398 A247399 A247400


KEYWORD

nonn,fini,full


AUTHOR

Elliott Line, Sep 16 2014


STATUS

approved



