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%I #39 Mar 02 2024 03:29:10
%S 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,
%T 26,35,31,31,34,35
%N Excess area when consecutive squares of sizes 1 to n are packed into the smallest possible rectangle.
%C Restricted to packings with the squares aligned with the sides of the rectangle.
%D R. K. Guy, Unsolved Problems in Geometry, Section D4, has information about several related problems.
%D R. M. Kurchan (editor), Puzzle Fun, Number 18 (December 1997), pp. 9-10.
%H Jean-François Alcover, <a href="/A081287/a081287.txt">Mathematica script (after E. Pegg and R. Korf)</a>
%H R. Ellard and Des MacHale, <a href="https://doi.org/10.1017/S0025557200003922">Packing Squares into Rectangles</a>, The Mathematical Gazette, Vol. 96, No. 535 (March 2012), 1-18.
%H Eric Huang and Richard E. Korf, <a href="http://search-conference.org/index.php/Main/SOCS09program?action=download&upname=SoCS09-13.pdf">New improvements in optimal rectangle packing</a>
%H Richard E. Korf, <a href="https://www.aaai.org/Library/ICAPS/2004/icaps04-019.php">Optimal Rectangle Packing: New Results</a>, ICAPS, 2004.
%H Ed Pegg Jr, <a href="http://www.maa.org/editorial/mathgames/mathgames_12_01_03.html">Packing squares</a>
%H E. Pegg and R. Korf, <a href="http://demonstrations.wolfram.com/TightlyPackedSquares/">Tightly Packed Squares</a>.
%F a(n) = A038666(n) - A000330(n). - _Pontus von Brömssen_, Mar 01 2024
%e Verified best rectangles > 5 are as follows:
%e 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
%e --------------------------------------------------------------------------------------
%e 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
%e 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
%e Visual representations are at the Tightly Packed Squares link.
%Y Cf. A000330, A038666, A369891.
%K nice,nonn,more
%O 1,4
%A _Ed Pegg Jr_, Mar 16 2003
%E Four extra terms computed by Korf, May 24 2005
%E More terms from _Ed Pegg Jr_, Feb 14 2008 and again Sep 16 2009
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