A081287 Excess area when consecutive squares of sizes 1 to n are packed into the smallest possible rectangle.

0, 1, 1, 5, 5, 8, 14, 6, 15, 20, 7, 17, 17, 20, 25, 16, 9, 30, 21, 20, 33, 27, 28, 28, 22, 29, 26, 35, 31, 31, 34, 35

Offset: 1

Author

Ed Pegg Jr, Mar 16 2003

Keywords

• nice
• nonn
• more

Comments

Restricted to packings with the squares aligned with the sides of the rectangle.

Examples

			Verified best rectangles > 5 are as follows:
 6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24  25 26 27  28  29  30  31  32
--------------------------------------------------------------------------------------
 9 11 14 15 15 19 23 22 23 23 28 39 31 47 34 38 39 64 56  43 70 74  63  81  51  95  85
11 14 15 20 27 27 29 38 45 55 54 46 69 53 85 88 98 68 88 129 89 94 123 106 186 110 135
Visual representations are at the Tightly Packed Squares link.
		

References

• R. K. Guy, Unsolved Problems in Geometry, Section D4, has information about several related problems.
• R. M. Kurchan (editor), Puzzle Fun, Number 18 (December 1997), pp. 9-10.

Links

• Jean-François Alcover, Mathematica script (after E. Pegg and R. Korf)
• R. Ellard and Des MacHale, Packing Squares into Rectangles, The Mathematical Gazette, Vol. 96, No. 535 (March 2012), 1-18.
• Eric Huang and Richard E. Korf, New improvements in optimal rectangle packing
• Richard E. Korf, Optimal Rectangle Packing: New Results, ICAPS, 2004.
• Ed Pegg Jr, Square Packing
• E. Pegg and R. Korf, Tightly Packed Squares.

Crossrefs

Cf. A000330, A038666, A369891.

Formula

a(n) = A038666(n) - A000330(n). - Pontus von Brömssen, Mar 01 2024

Extensions

Four extra terms computed by Korf, May 24 2005

More terms from Ed Pegg Jr, Feb 14 2008 and again Sep 16 2009