A279596 - OEIS

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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 (list; graph; refs; listen; history; text; internal format)

	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`

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Last modified July 16 21:35 EDT 2021. Contains 346077 sequences. (Running on oeis4.)