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.

0, 1, 1, 3, 2, 4, 3, 3, 4

Offset

1,4

Comments

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.

Links

Table of n, a(n) for n=1..9.

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.

Formula

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.

Example

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

Crossrefs

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

Sequence in context: A025509 A130079 A350929 * A247190 A243289 A134559

Adjacent sequences: A365233 A365234 A365235 * A365237 A365238 A365239

Keyword

nonn,more

Author

Tamas Sandor Nagy and Thomas Scheuerle, Sep 25 2023

Extensions

Edited by Peter Munn, Oct 04 2023

Status

approved