A081287
Excess area when consecutive squares of sizes 1 to n are packed into the smallest possible rectangle.
Original entry on oeis.org
0, 1, 1, 5, 5, 8, 14, 6, 15, 20, 7, 17, 17, 20, 25, 16, 9, 30, 21, 20, 33, 27, 28, 28, 22, 29, 26, 35, 31, 31, 34, 35
Offset: 1
Verified best rectangles > 5 are as follows:
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
--------------------------------------------------------------------------------------
9 11 14 15 15 19 23 22 23 23 28 39 31 47 34 38 39 64 56 43 70 74 63 81 51 95 85
11 14 15 20 27 27 29 38 45 55 54 46 69 53 85 88 98 68 88 129 89 94 123 106 186 110 135
Visual representations are at the Tightly Packed Squares link.
- R. K. Guy, Unsolved Problems in Geometry, Section D4, has information about several related problems.
- R. M. Kurchan (editor), Puzzle Fun, Number 18 (December 1997), pp. 9-10.
- Jean-François Alcover, Mathematica script (after E. Pegg and R. Korf)
- R. Ellard and Des MacHale, Packing Squares into Rectangles, The Mathematical Gazette, Vol. 96, No. 535 (March 2012), 1-18.
- Eric Huang and Richard E. Korf, New improvements in optimal rectangle packing
- Richard E. Korf, Optimal Rectangle Packing: New Results, ICAPS, 2004.
- Ed Pegg Jr, Square Packing
- E. Pegg and R. Korf, Tightly Packed Squares.
Four extra terms computed by Korf, May 24 2005
More terms from
Ed Pegg Jr, Feb 14 2008 and again Sep 16 2009
A365236
a(n) is the least number of integer-sided squares that can be packed together with the n squares 1 X 1, 2 X 2, ..., n X n to fill out a rectangle.
Original entry on oeis.org
0, 1, 1, 3, 2, 4, 3, 3, 4
Offset: 1
Compositions of rectangles that satisfy the minimal number of augmenting squares for n. Where more than one minimal composition exists for a given n, the table shows a single example. In the table body, the numbers include both the specific mandatory and augmenting squares. a(n) is the total number of squares in the rectangle minus n.
| 1^2 2^2 3^2 4^2 5^2 6^2 7^2 8^2 9^2 10^2 | Total
----------------------------------------------------------------------------
a(1) = 0 | 1 | 1
a(2) = 1 | 2 1 | 3
a(3) = 1 | 2 1 1 | 4
a(4) = 3 | 2 1 2 2 | 7
a(5) = 2 | 2 1 1 2 1 | 7
a(6) = 4 | 2 1 3 2 1 1 | 10
a(7) = 3 | 1 1 1 3 1 2 1 | 10
a(8) = 3 | 3 2 1 1 1 1 1 1 | 11
a(9) = 4 | 2 2 2 2 1 1 1 1 1 | 13
- Tamas Sandor Nagy, Examples for a(1) - a(4).
- Tamas Sandor Nagy, Example for a(5).
- Tamas Sandor Nagy, Example for a(6).
- Tamas Sandor Nagy, Original upper bound examples for a(7) with 5 augmenting squares and a(8) with 6 augmenting squares.
- Tamas Sandor Nagy, Example of a conjectured solution for a(10) with 5 augmenting squares, found by Peter Munn.
- Thomas Scheuerle, Example for a(6) with smallest possible area.
- Thomas Scheuerle, Example for a(7).
- Thomas Scheuerle, Example for a(8).
- Thomas Scheuerle, Example for a(9).
- Thomas Scheuerle, Original upper bound example for a(10) with 6 augmenting squares.
- Thomas Scheuerle, Example for a(11) = 4 this is at the same time also a conjectured solution for a(10) = 5.
Showing 1-2 of 2 results.
Comments