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

1,4

Comments

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

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

Table of n, a(n) for n=1..32.

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, Packing squares

E. Pegg and R. Korf, Tightly Packed Squares.

Formula

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

Example

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.

Crossrefs

Cf. A000330, A038666, A369891.

Sequence in context: A107623 A218333 A212533 * A303715 A377285 A204188

Adjacent sequences: A081284 A081285 A081286 * A081288 A081289 A081290

Keyword

nice,nonn,more

Author

Ed Pegg Jr, Mar 16 2003

Extensions

Four extra terms computed by Korf, May 24 2005

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

Status

approved