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.

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, 8, 8, 8, 8, 8, 8, 8, 8, 9

Offset

3,1

Comments

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

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.

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).

Links

Table of n, a(n) for n=3..45.

Robert Gerbicz, Optimal tilings for n = 3..45

Mersenneforum.org puzzles, Mondrian art puzzles.

Ed Pegg Jr, Mondrian Art Problem.

Ed Pegg Jr, Mondrian Art Problem Upper Bound for defect.

Example

The 9 X 9 square can be divided into non-translatable rectangles with
aaaaaaaab
ddddddeeb
fggghheeb
fggghheeb
fiiihheeb
fiiijjjjb
fiiijjjjb
fkkkkkkkb
ccccccccc

Crossrefs

Cf. A276523, A278970, A279848.

Sequence in context: A111393 A323665 A062537 * A224458 A097688 A262685

Adjacent sequences: A279593 A279594 A279595 * A279597 A279598 A279599

Keyword

hard,more,nonn

Author

Ed Pegg Jr, Dec 15 2016

Extensions

Moved terms to A279848, expanded best values known

a(28)-a(45) from Robert Gerbicz, Jan 01 2017

Status

approved