cp's OEIS Frontend

A279596 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.

%I A279596 #18 Feb 11 2020 02:14:09
%S A279596 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 A279596 8,8,8,8,8,8,8,8,9
%N A279596 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 A279596 Similar to the Mondrian Art sequence (A276523), but allowing repetition of rectangles with different orientations.
%C A279596 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 A279596 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 A279596 Robert Gerbicz, <a href="/A279596/a279596.txt">Optimal tilings for n = 3..45</a>
%H A279596 Mersenneforum.org puzzles, <a href="http://mersenneforum.org/showthread.php?t=21775">Mondrian art puzzles</a>.
%H A279596 Ed Pegg Jr, <a href="http://demonstrations.wolfram.com/MondrianArtProblem/">Mondrian Art Problem</a>.
%H A279596 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 A279596 The 9 X 9 square can be divided into non-translatable rectangles with
%e A279596 aaaaaaaab
%e A279596 ddddddeeb
%e A279596 fggghheeb
%e A279596 fggghheeb
%e A279596 fiiihheeb
%e A279596 fiiijjjjb
%e A279596 fiiijjjjb
%e A279596 fkkkkkkkb
%e A279596 ccccccccc
%Y A279596 Cf. A276523, A278970, A279848.
%K A279596 hard,more,nonn
%O A279596 3,1
%A A279596 _Ed Pegg Jr_, Dec 15 2016
%E A279596 Moved terms to A279848, expanded best values known
%E A279596 a(28)-a(45) from _Robert Gerbicz_, Jan 01 2017