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 A276523 Partition an n X n square into multiple non-congruent integer-sided rectangles. a(n) is the least possible difference between the largest and smallest area. 4
 2, 4, 4, 5, 5, 6, 6, 8, 6, 7, 8, 6, 8, 8, 8, 8, 8, 9, 9, 9, 8, 9, 10, 9, 10, 9, 9, 11, 11, 10, 12, 12, 11, 12, 11, 10, 11, 12, 13, 12, 12, 12, 13, 13, 12, 14, 12, 13, 14, 13, 14, 15, 14, 14, 15, 15, 14, 15, 15, 14, 15, 15, 15 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,1 COMMENTS Developed as the Mondrian Art Puzzle. The rectangles can be similar, though. - Daniel Forgues, Nov 22 2016 That is, there can be a 1x2 rectangle and a 2x4 rectangle (these are similar), but there can't be two 1x2 rectangles (these are congruent). - Michael B. Porter, Oct 13 2018 Upper bounds for a(n) are n if n is odd, and min(2*n, 4 * a(n/2)) if n is even. - Roderick MacPhee, Nov 28 2016 An upper bound seems to be ceiling(n/log(n))+3, or A050501+3. See A278970. Holds to at least a(96). - Ed Pegg Jr, Dec 02 2016 Best known values for a(58)-a(96) as follows: 16, 15, 18, 15, 16, 18, 15, 18, 16, 18, 19, 18, 19, 18, 20, 20, 20, 20, 19, 20, 21, 21, 20, 21, 20, 20, 21, 22, 18, 22, 20, 22, 24, 23, 22, 22, 24, 24, 24 LINKS Table of n, a(n) for n=3..65. Michel Gaillard, Optimal tilings for n=58..65 Robert Gerbicz, Optimal tilings for n=3..57 Gordon Hamilton, Mondrian Art Puzzles (2015). Gordon Hamilton and Brady Haran, Mondrian Puzzle, Numberphile video (2016) Mersenneforum.org puzzles, Mondrian art puzzles Cooper O'Kuhn, The Mondrian Puzzle: A Connection to Number Theory, arXiv:1810.04585 [math.CO], 2018. Cooper O'Kuhn and Todd Fellman, The Mondrian Puzzle: A Bound Concerning the M(n)=0 Case, arXiv:2006.12547 [math.NT], 2020. See also Integers (2021) Vol. 21, #A37. Ed Pegg Jr, Mondrian Art Problem. EXAMPLE A size-11 square can be divided into 3 X 4, 2 X 6, 2 X 7, 3 X 5, 4 X 4, 2 X 8, 2 X 9, and 3 X 6 rectangles. 18 - 12 = 6, the minimal area range. The 14 X 14 square can be divided into non-congruent rectangles of area 30 to 36: aaaaaaaaaabbbb aaaaaaaaaabbbb aaaaaaaaaabbbb cccdddddddbbbb cccdddddddbbbb cccdddddddbbbb cccdddddddbbbb cccdddddddbbbb ccceeeeeffffff ccceeeeeffffff ccceeeeeffffff ccceeeeeffffff ccceeeeeffffff ccceeeeeffffff CROSSREFS Cf. A050501, A278970, A279596. Sequence in context: A036437 A053306 A108422 * A244320 A084616 A196259 Adjacent sequences: A276520 A276521 A276522 * A276524 A276525 A276526 KEYWORD nonn,hard,more AUTHOR Ed Pegg Jr, Nov 15 2016 EXTENSIONS Bruce Norskog corrected a(18), and a recheck by Pegg corrected a(15) and a(19). - Charles R Greathouse IV, Nov 28 2016 Correction of a(14), a(16), a(23) and new terms a(25)-a(28) from Robert Gerbicz, Nov 28 2016 a(29)-a(44) from Robert Gerbicz, Dec 02 2016 a(45)-a(47) from Robert Gerbicz added, as well as best known values to a(96). Correction of a(45), a(46) and new terms a(48)-a(57) from Robert Gerbicz, Dec 27 2016 a(58)-a(65) from Michel Gaillard, Oct 23 2020 STATUS approved

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