This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A014530 #43 Mar 13 2025 12:53:09 %S A014530 2,4,6,7,8,9,11,15,16,17,18,19,24,25,27,29,33,35,37,42,50 %N A014530 List of sizes of squares occurring in lowest order example of a perfect squared square. %C A014530 A squared rectangle (which may be a square) is a rectangle dissected into a finite number, two or more, of squares. If no two of these squares have the same size the squared rectangle is perfect. A squared rectangle is simple if it does not contain a smaller squared rectangle, and compound if it does. The order of a squared rectangle is the number of constituent squares. Duijvestijn's perfect square of lowest order (21) is simple. The lowest order of a compound perfect square is 24. [_Geoffrey H. Morley_, Oct 17 2012] %C A014530 See the MathWorld link for an explanation of Bouwkamp code. The Bouwkamp code for the squaring is (50,35,27)(8,19)(15,17,11)(6,24)(29,25,9,2)(7,18)(16)(42)(4,37)(33). [_Geoffrey H. Morley_, Oct 18 2012] %D A014530 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences. Academic Press, San Diego, 1995, Fig. M4482. %D A014530 I. Stewart, Squaring the Square, Scientific Amer., 277, July 1997, pp. 94-96. %H A014530 Stuart E. Anderson, <a href="http://www.squaring.net/downloads/downloads.html#pss">Catalogues of Perfect Squared Squares</a> %H A014530 C. J. Bouwkamp and A. J. W. Duijvestijn, <a href="http://alexandria.tue.nl/repository/books/391207.pdf">Catalogue of Simple Perfect Squared Squares of orders 21 through 25</a>, EUT Report 92-WSK-03, Eindhoven University of Technology, Eindhoven, The Netherlands, November 1992. %H A014530 C. J. Bouwkamp and A. J. W. Duijvestijn, <a href="http://alexandria.tue.nl/repository/books/430534.pdf">Album of Simple Perfect Squared Squares of order 26</a>, EUT Report 94-WSK-02, Eindhoven University of Technology, Eindhoven, The Netherlands, December 1994. %H A014530 A. J. W. Duijvestijn, <a href="http://doc.utwente.nl/68433/1/Duijvestijn78simple.pdf">Simple perfect square of lowest order</a>, J. Combin. Theory Ser. B 25 (1978), 240-243. %H A014530 Gergely Földvári, <a href="/A014530/a014530.jpg">Photo of my artwork (2022) depicting the lowest order perfect squared square using 21 distinct colors</a> %H A014530 N. D. Kazarinoff and R. Weitzenkamp, <a href="http://deepblue.lib.umich.edu/bitstream/2027.42/33904/1/0000169.pdf">On the existence of compound perfect squared squares of small order</a>, J. Combin. Theory Ser. B 14 (1973).163-179. [A compound perfect squared square must contain at least 22 subsquares.] %H A014530 Trinity College Mathematical Society, <a href="https://www.srcf.ucam.org/tms/about-the-tms/the-squared-square/">The Squared Square</a> %H A014530 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PerfectSquareDissection.html">Perfect Square Dissection</a> %H A014530 <a href="/index/Sq#squared_squares">Index entries for squared squares</a> %e A014530 Example from _Rainer Rosenthal_, Mar 25 2021: (Start) %e A014530 . %e A014530 Terms | 2 4 6 7 8 9 11 15 16 17 18 19 24 25 27 29 33 35 37 42 50 %e A014530 ------------------------------------------------------------------------- %e A014530 | <-- sort selected groups %e A014530 ------------------------------------------------------------------------- %e A014530 (50,35,27) | . . . . . . . . . . . . . . 27 . . 35 . . 50 %e A014530 (8,19) | . . . . 8 . . . . . . 19 . . . . . . %e A014530 (15,17,11) | . . . . . 11 15 . 17 . . . . . . . %e A014530 (6,24) | . . 6 . . . . 24 . . . . . %e A014530 (29,25,9,2)| 2 . . 9 . . 25 29 . . . %e A014530 (7,18) | . 7 . 18 . . . %e A014530 (16) | . 16 . . . %e A014530 (42) | . . . 42 %e A014530 (4,37) | 4 . 37 %e A014530 (33) | 33 %e A014530 _________________________________________________________________________ %e A014530 Groups of terms selected and sorted for the Bouwkamp piling %e A014530 . %e A014530 The Bouwkamp code says how to pile up the squares in order to tile the square with side length 50 + 35 + 27 = 112. The procedure is beautifully animated in "Eric Weisstein's World of Mathematics" entry - see link. %e A014530 (End) %Y A014530 Cf. A002839, A002962, A002881, A342558 (related by the analogy between square tilings and resistor networks). %K A014530 nonn,fini,full %O A014530 1,1 %A A014530 _N. J. A. Sloane_ %E A014530 'Simple' removed from definition by _Geoffrey H. Morley_, Oct 17 2012