A279596 - OEIS

%I #18 Feb 11 2020 02:14:09

%S 2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,6,4,6,6,6,6,6,6,7,7,7,8,7,6,6,8,7,8,

%T 8,8,8,8,8,8,8,8,9

%N Partition an n X n square into multiple integer-sided rectangles where no one is a translation of any other; a(n) is the least possible difference between the largest and smallest area.

%C Similar to the Mondrian Art sequence (A276523), but allowing repetition of rectangles with different orientations.

%C Proved optimal to a(45) by R. Gerbicz.  Best values known for a(46)-a(96): 10, 12, 11, 12, 12, 8, 12, 12, 13, 12, 12, 14, 14, 15, 12, 15, 14, 15, 14, 16, 16, 15, 16, 16, 16, 17, 16, 17, 14, 17, 18, 16, 18, 16, 18, 15, 16, 18, 18, 16, 18, 17, 19, 20, 17, 17, 21, 20, 20, 21, 22.

%C Seems to be bounded above by ceiling(n/log(n)). The currently verified distances from this bound are 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 3, 1, 2, 2, 2, 2, 2, 2 (A279848).

%H Robert Gerbicz, <a href="/A279596/a279596.txt">Optimal tilings for n = 3..45</a>

%H Mersenneforum.org puzzles, <a href="http://mersenneforum.org/showthread.php?t=21775">Mondrian art puzzles</a>.

%H Ed Pegg Jr, <a href="http://demonstrations.wolfram.com/MondrianArtProblem/">Mondrian Art Problem</a>.

%H Ed Pegg Jr, <a href="http://math.stackexchange.com/questions/2041189/mondrian-art-problem-upper-bound-for-defect">Mondrian Art Problem Upper Bound for defect</a>.

%e The 9 X 9 square can be divided into non-translatable rectangles with

%e aaaaaaaab

%e ddddddeeb

%e fggghheeb

%e fggghheeb

%e fiiihheeb

%e fiiijjjjb

%e fiiijjjjb

%e fkkkkkkkb

%e ccccccccc

%Y Cf. A276523, A278970, A279848.

%K hard,more,nonn

%O 3,1

%A _Ed Pegg Jr_, Dec 15 2016

%E Moved terms to A279848, expanded best values known

%E a(28)-a(45) from _Robert Gerbicz_, Jan 01 2017