Search: id:A005842
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%I A005842 M2401
%S A005842 1,3,5,7,9,11,13,15,18,21,24,27,30,33,36,39,43,47,50,54,58,62,66,71,75,
%T A005842 80,84,89,93,98,103,108,113,118,123,128,133,139,144,150,155,161,166,172,
%U A005842 178,184,190,196,202,208,214,221,227,233,240,246
%N A005842 a(n) = minimal integer m such that an m X m square contains all squares 
               of sides 1, ..., n (some values are only conjectures).
%C A005842 The entries a(1)=1, a(2)=3, a(8)=15, a(15)=36, a(16)=39, a(17)=43, a(19)=50, 
               a(20)=54, a(21)=58, a(22)=62, a(23)=66, a(25)=75, a(27)=84, a(29)=93, 
               a(30)=98, a(31)=103, a(35)=123, a(36)=128, a(37)=133, a(39)=144, 
               a(41)=155, a(43)=166, a(44)=172,
%C A005842 a(45)=178, a(46)=184, a(49)=202, a(50)=208, a(51)=214, a(54)=233, a(56)=246, 
               a(62)=286, a(63)=293, a(64)=300, a(65)=307, a(76)=387, a(90)=498 
               and a(94)=531 all meet the lower bound in A092137 and are therefore 
               correct. [Comment updated by Stuart Anderson (stuart(AT)squaring.net), 
               Jan 05 2008]
%C A005842 The values a(3)=5, a(4)=7, a(5)=9, a(6)=11, a(7)=13 are correct, since 
               it can be shown that the size of smallest square that contains two 
               squares of sides a and b is a+b. As a consequence A060747(n)=2n-1 
               is a lower bound for a(n). - Dan Dima (dimad72(AT)gmail.com), Jan 
               28 2007
%C A005842 This problem has a long history and the people credited in Ed Pegg's 
               description (including myself) are not the first to find the packings 
               that he attributes. The values given are upper bounds and conjectured 
               values and have been proved except those for n=26, 28, 32, 33, 34, 
               38, 40, 42, 47, 48, 52, 53 and 55, where they remain probable. Not 
               only is 2n-1 a lower bound (sharp for n<=8), so is 3n-9 because of 
               the 3rd, 4th and 5th smallest squares (sharp for 8n<=n<=16). - Erich 
               Friedman (efriedma(AT)stetson.edu), Jul 17 2007
%C A005842 Comment from Erich Friedman, May 27 2009: Simonis, H. and O'Sullivan 
               show that a(26) = 80. The first first unknown value is now a(28), 
               though the answer is almost surely 89.
%D A005842 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005842 H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, 
               D5.
%D A005842 M. Gardner, Mathematical Carnival. Random House, NY, 1977, p. 147.
%D A005842 Simonis, H. and O'Sullivan, B., Search Strategies for Rectangle Packing, 
               in Proceedings of the 14th international conference on Principles 
               and Practice of Constraint Programming, Springer-Verlag Berlin, Heidelberg, 
               2008, pp. 52-66.
%H A005842 Erich Friedman, <a href="http://www.stetson.edu/~efriedma/mathmagic/1099.html">
               Math Magic</a>.
%H A005842 Shigeyoshi Kamakura, <a href="a005842_23.gif">Illustration for n = 23</
               a>
%H A005842 Minami Kawasaki, <a href="a005842_19.gif">Illustration for n = 19</a>
%H A005842 Minami Kawasaki, <a href="a005842_22.gif">Illustration for n = 22</a>
%H A005842 Minami Kawasaki, <a href="http://www.geocities.co.jp/Berkeley-Labo/6317/
               seqsqr_00.htm">Catalogue of known solutions</a>
%H A005842 Ed Pegg Jr, <a href="a005842_25.gif">Illustration for n = 25, 27, 29, 
               30, 31, 35</a>
%H A005842 Ed Pegg Jr, <a href="a005842_37.gif">Illustration for n = 37</a>
%H A005842 R.E. Korf, <a href="https://www.aaai.org/Papers/ICAPS/2004/ICAPS04-019.pdf">
               Optimal Rectangle Packing: New Results</a>, Proceedings of the International 
               Conference on Automated Planning and Scheduling (ICAPS04), Whistler, 
               British Columbia, June 2004, pp. 142-149. [From Rob Pratt (Rob.Pratt(AT)sas.com), 
               Jun 10 2009]
%e A005842 75^2 - Sum[n^2, {n, 1, 25}] == 100 Solvable - Erich Friedman
%e A005842 79^2 - Sum[n^2, {n, 1, 26}] == 40 Highly unlikely
%e A005842 84^2 - Sum[n^2, {n, 1, 27}] == 126 Solvable - Robert Reid
%e A005842 88^2 - Sum[n^2, {n, 1, 28}] == 30 Highly unlikely
%e A005842 93^2 - Sum[n^2, {n, 1, 29}] == 94 Solvable - Ed Pegg Jr
%e A005842 98^2 - Sum[n^2, {n, 1, 30}] == 149 Solvable - Robert Reid
%e A005842 103^2 - Sum[n^2, {n, 1, 31}] == 193 Solvable - RR and EF
%e A005842 107^2 - Sum[n^2, {n, 1, 32}] == 9 Very highly unlikely
%e A005842 112^2 - Sum[n^2, {n, 1, 33}] == 15 Very highly unlikely
%e A005842 117^2 - Sum[n^2, {n, 1, 34}] == 4 Very highly unlikely
%e A005842 123^2 - Sum[n^2, {n, 1, 35}] == 219 Solvable - Erich Friedman
%e A005842 128^2 - Sum[n^2, {n, 1, 36}] == 178 Likely hand-solvable
%Y A005842 Cf. A092137 (lower bound).
%Y A005842 Sequence in context: A137507 A061808 A143450 this_sequence A081110 A143449 
               A033034
%Y A005842 Adjacent sequences: A005839 A005840 A005841 this_sequence A005843 A005844 
               A005845
%K A005842 nonn
%O A005842 1,2
%A A005842 N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
%E A005842 a(25)-a(36) reported by Ed Pegg Jr. (ed(AT)mathpuzzle.com), Jan 04 2004
%E A005842 a(23) = 66 from Shigeyoshi Kamakura, Mar 29 2004
%E A005842 a(37) = 133, solved independently by Robert Reid and Berend Jan van der 
               Zwaag, Mar 30 2004
%E A005842 It is not clear which of these entries have been proved to be correct. 
               At the present time they should all be regarded as conjectures unless 
               indicated otherwise. N. J. A. Sloane (njas(AT)research.att.com), 
               Mar 30 2004. Entry revised Apr 06 2004.
%E A005842 More terms from Erich Friedman (efriedma(AT)stetson.edu), Jul 17 2007
%E A005842 Corrected a comment on a lower bound. - Jon E. Schoenfield (jonscho(AT)hiwaay.net), 
               Nov 19 2008

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