A247397 Numbers n such that when n unit-diameter circles are arranged non-overlapping in the plane, and those circles are then enclosed in a rectangle, the area of the rectangle must be at least n.

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

Offset

1,2

Comments

For any number that does not appear on this list, there exists an arrangement of that number of unit-diameter circles that can be enclosed in a rectangle with area of less than 1 square unit per circle.

Any number of unit-diameter 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 unit-diameter 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 unit-diameter 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