A219158 Minimum number of integer-sided squares needed to tile an m X n rectangle.

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

Offset: 1

Author

David Radcliffe, Nov 12 2012

Keywords

• nonn
• tabl

Comments

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

Examples

			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.
Triangle begins:
   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, 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, 6, 4, 3, 6, 2, 6, 3, 4, 5, 7, 1;
  13, 8, 7, 7, 6, 6, 6, 6, 7, 7, 6, 7, 1;
  14, 7, 7, 5, 7, 5, 2, 5, 7, 5, 7, 5, 7, 1;
  15, 9, 5, 7, 3, 4, 8, 8, 4, 3, 7, 5, 8, 7, 1;
		

Links

• Massimo Ortolano, Table of n, a(n) for n = 1..75466, rows 1..388 of triangle, flattened. Corrected version provided by Qizheng He.
• Gary Antonick, Matt Enlow's Rectangle Division Puzzle, The New York Times, June 15, 2015.
• Bertram Felgenhauer, Filling rectangles with integer-sided squares
• Richard J. Kenyon, Tiling a rectangle with the fewest squares, Combin. Theory Ser. A 76 (1996), no. 2, 272-291.
• M. Ortolano, M. Abrate, and L. Callegaro, On the synthesis of Quantum Hall Array Resistance Standards, arXiv preprint arXiv:1311.0756 [physics.ins-det], 2013.
• Mark Walters, Rectangles as sums of squares, Discrete Math. 309 (2009), no. 9, 2913-2921.

Crossrefs

First 19 terms agree with A049834.

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

Cf. A113881, A338861.