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A279317 Minimal number of squares in a dissection of an (n) X (n+1) oblong into squares.

%I A279317 #59 Jan 03 2021 15:53:42
%S A279317 2,3,4,5,5,5,7,7,6,6,7,7,7,7,7,8,8,7,9,8,8,8,8,8,8,8,8,9,9,8,9,9,9,8,
%T A279317 9,9,9,9,9,9,9,9,9,10,9,9,10,9,10,9,10,10,10,10,9,10,10,10,10,10,10,
%U A279317 10,10,10,10,10,10,10,11,10,10,10,10,10,10,10,11,11,10,10,10,10,10,10,11,10,11,10,11,10,11,11,11,11,10,11,11,11,11,11,11,11,11,10,11,11,11,11,11,11,11,11,12
%N A279317 Minimal number of squares in a dissection of an (n) X (n+1) oblong into squares.
%C A279317 This is very close to b(n) = round(n^(1/3)) + 6. b(18)-a(18) = 2. b(387)-a(387) = 0. All b(n)-a(n) terms in between these points are -1, 0, 1.
%C A279317 Bouwkamp codes of dissections that are believed to be optimal follow.
%C A279317 10 105 104 60 45 19 26 44 16 12 7 33 28
%C A279317 11 177 176 99 78 21 57 77 43 16 41 34 9 25
%C A279317 12 308 307 165 143 22 67 54 142 45 13 41 97 28 69
%C A279317 13 552 551 312 240 44 60 136 28 16 76 239 101 37 175 138
%C A279317 14 970 969 546 424 172 252 423 73 50 23 119 80 96 39 293 254
%C A279317 15 1699 1698 951 748 307 441 747 127 77 50 27 200 134 177 66 509 443
%C A279317 16 2926 2925 1633 1293 213 299 781 127 86 41 344 1292 509 206 138 68 851 783
%C A279317 17 5211 5210 2846 2365 571 518 1276 2364 392 90 53 465 302 412 694 584 293 1569 1278
%C A279317 18 8731 8730 4741 3990 751 1195 2044 3989 1059 444 790 849 884 175 709 256 197 2696 453 2046
%C A279317 19 15131 15130 8169 6962 2415 4547 6961 1208 1943 1680 263 965 452 1504 702 1378 3621 802 865 2306 2243
%C A279317 20 25679 25678 13719 11960 1456 1866 2626 6012 303 743 410 11959 1623 440 1516 760 1183 3386 4322 1692 7706 6014
%C A279317 21 49583 49582 27252 22331 4763 5036 12532 158 4332 273 22330 5080 5309 906 2176 1250 4716 1270 4372 2187 3446 14719 12534
%H A279317 Ed Pegg Jr, <a href="/A279317/b279317.txt">Table of n, a(n) for n = 1..387</a>
%H A279317 S. Anderson, <a href="http://www.squaring.net/sq/sr/spsr/spsr.html">Catalogues of Simple Perfect Squared Rectangles (SPSR)</a>
%H A279317 B. Felgenhauer, <a href="http://int-e.eu/~bf3/squares/">Filling Rectangles with Integer-Sided Squares</a>.
%H A279317 Ed Pegg Jr, <a href="http://demonstrations.wolfram.com/MinimallySquaredRectangles/">Minimally Squared Rectangles</a>.
%H A279317 Ed Pegg Jr on StackExchange, <a href="http://math.stackexchange.com/questions/2057290/oblongs-into-minimal-squares">Oblongs into minimal squares</a>, Dec 13 2016.
%e A279317 Oblong 18 X 19 uses 7 squares of size 3, 5, 5, 7, 7, 8, 11.
%e A279317 Oblong 34 X 35 uses 8 squares of size 4, 7, 9, 9, 11, 15, 16, 19.
%e A279317 Oblong 55 X 56 uses 9 squares of size 5, 9, 12, 12, 14, 19, 23, 24, 32.
%e A279317 Oblong 104 X 105 uses 10 squares of size 7, 12, 16, 19, 26, 28, 33, 44, 45, 60.
%e A279317 From _Peter Kagey_, Dec 13 2016: (Start)
%e A279317 An example of the a(10) = 6 squares that can dissect a 10 X 11 oblong:
%e A279317   +-------+-----------+
%e A279317   |       |           |
%e A279317   |   4   |           |
%e A279317   |       |     6     |
%e A279317   +---+---+           |
%e A279317   | 2 | 2 |           |
%e A279317   +---+---+-+---------+
%e A279317   |         |         |
%e A279317   |    5    |    5    |
%e A279317   |         |         |
%e A279317   |         |         |
%e A279317   +---------+---------+
%e A279317 (End)
%Y A279317 Cf. A005670, A339548.
%K A279317 nonn,hard
%O A279317 1,1
%A A279317 _Ed Pegg Jr_, Dec 09 2016
%E A279317 Corrected term 351 and extended to n=387 by _Ed Pegg Jr_, Oct 31 2018