%I #71 Nov 28 2023 16:00:07

%S 1,2,1,3,3,1,4,2,4,1,5,4,4,5,1,6,3,2,3,5,1,7,5,5,5,5,5,1,8,4,5,2,5,4,

%T 7,1,9,6,3,6,6,3,6,7,1,10,5,6,4,2,4,6,5,6,1,11,7,6,6,6,6,6,6,7,6,1,12,

%U 6,4,3,6,2,6,3,4,5,7,1,13,8,7,7,6,6,6,6,7,7,6,7,1

%N Minimum number of integer-sided squares needed to tile an m X n rectangle.

%C Triangular array read by rows. m=1,2,...,n; n=1,2,3,...

%H Massimo Ortolano, <a href="/A219158/b219158.txt">Table of n, a(n) for n = 1..75466</a>, rows 1..388 of triangle, flattened. Corrected version provided by Qizheng He.

%H Gary Antonick, <a href="https://wordplay.blogs.nytimes.com/2015/06/15/enlow-2/">Matt Enlow's Rectangle Division Puzzle</a>, The New York Times, June 15, 2015.

%H Bertram Felgenhauer, <a href="http://int-e.eu/~bf3/squares/">Filling rectangles with integer-sided squares</a>

%H Richard J. Kenyon, <a href="http://dx.doi.org/10.1006/jcta.1996.0104">Tiling a rectangle with the fewest squares</a>, Combin. Theory Ser. A 76 (1996), no. 2, 272-291.

%H M. Ortolano, M. Abrate, and L. Callegaro, <a href="http://arxiv.org/abs/1311.0756">On the synthesis of Quantum Hall Array Resistance Standards</a>, arXiv preprint arXiv:1311.0756 [physics.ins-det], 2013.

%H Mark Walters, <a href="http://dx.doi.org/10.1016/j.disc.2008.07.028">Rectangles as sums of squares</a>, Discrete Math. 309 (2009), no. 9, 2913-2921.

%e T(6,5) = 5 because a 6 X 5 rectangle can be subdivided into two 3 X 3 squares and three 2 X 2 squares.

%e Triangle begins:

%e 1;

%e 2, 1;

%e 3, 3, 1;

%e 4, 2, 4, 1;

%e 5, 4, 4, 5, 1;

%e 6, 3, 2, 3, 5, 1;

%e 7, 5, 5, 5, 5, 5, 1;

%e 8, 4, 5, 2, 5, 4, 7, 1;

%e 9, 6, 3, 6, 6, 3, 6, 7, 1;

%e 10, 5, 6, 4, 2, 4, 6, 5, 6, 1;

%e 11, 7, 6, 6, 6, 6, 6, 6, 7, 6, 1;

%e 12, 6, 4, 3, 6, 2, 6, 3, 4, 5, 7, 1;

%e 13, 8, 7, 7, 6, 6, 6, 6, 7, 7, 6, 7, 1;

%e 14, 7, 7, 5, 7, 5, 2, 5, 7, 5, 7, 5, 7, 1;

%e 15, 9, 5, 7, 3, 4, 8, 8, 4, 3, 7, 5, 8, 7, 1;

%Y First 19 terms agree with A049834.

%Y Columns m=1..10 give A001477, A030451, A226576, A226577, A226578, A226579, A226580, A226581, A226582, A226583.

%Y Cf. A113881, A338861.

%K nonn,tabl

%O 1,2

%A _David Radcliffe_, Nov 12 2012