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

Author

Elliott Line, Sep 16 2014

Keywords

• nonn
• fini
• full

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.

Examples

			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.
		

Links

• Eckard Specht, Densest known packings of a given number of circles in a rectangle