A081287 -id:A081287 - cp's OEIS frontend

A038666 Minimum area rectangle into which squares of sizes 1, 2, 3, ... n can be packed.

1, 6, 15, 35, 60, 99, 154, 210, 300, 405, 513, 667, 836, 1035, 1265, 1512, 1794, 2139, 2491, 2890, 3344, 3822, 4352, 4928, 5547, 6230, 6956, 7749, 8586, 9486, 10450, 11475

Offset: 1

Erich Friedman

R. M. Kurchan (editor), Puzzle Fun, Number 18 (December 1997), pp. 9-10.
R. M. Kurchan (editor), Solutions of Puzzle Fun 18, Puzzle Fun, Number 22 (2000), pp. 8-10.

Jean-François Alcover, Mathematica script (after E. Pegg and R. Korf)
R. Ellard and D. MacHale, Packing Squares into Rectangles, The Mathematical Gazette, Vol. 96, No. 535 (March 2012), 1-18.
E. Huang and R. E. Korf, New Improvements in Optimal Rectangle Packing, IJCAI-09: Proceedings of the 21st International Joint Conference on Artificial Intelligence, AAAI Press, 2009, pages 511-516. (Table 1 incorrectly lists the 95 X 110 minimal rectangle for n = 31 as 91 X 110.)
Ed Pegg, Jr., Illustration of 17th term

Cf. A000330, A081287.

a(n) = A000330(n) + A081287(n). - Pontus von Brömssen, Mar 01 2024

Corrected and extended by William Rex Marshall, Mar 23 2002 and Aug 29 2002

a(22)-a(32) from Korf, communicated by William Rex Marshall, May 03 2012

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

Tamas Sandor Nagy and Thomas Scheuerle, Sep 25 2023

Warning: several terms are provisional as their intended verification effectively assumed the augmenting squares were not larger than n X n. - Peter Munn, Oct 02 2023
The definition does not exclude squares larger than n X n.
Terms for n < 10 were verified by the use of a program.
a(10) <= 5.

			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.

Cf. A000290, A005670, A005842, A038666, A081287.

a(n) <= 1 + Sum_{k = 1 .. ceiling((n - 1)/2)} (n + (1 - k)*floor(n/k) - 2). This upper bound corresponds to placing the squares with length n up to n - floor((n - 1)/2) all in one row. The remaining mandatory squares will then fit naturally into the rectangle n X (1/2)*(2*n - ceiling((n - 1)/2))*(ceiling((n - 1)/2) + 1).

a(n) > a(n - 1) - 2.

Edited by Peter Munn, Oct 04 2023

cp's OEIS Frontend

A318931 a(n) = A081287(n) - n.

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A038666 Minimum area rectangle into which squares of sizes 1, 2, 3, ... n can be packed.

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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.

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