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%I #120 Oct 04 2023 12:33:42
%S 0,1,1,3,2,4,3,3,4
%N 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.
%C 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
%C The definition does not exclude squares larger than n X n.
%C Terms for n < 10 were verified by the use of a program.
%C a(10) <= 5.
%H Tamas Sandor Nagy, <a href="/A365236/a365236.png">Examples for a(1) - a(4)</a>.
%H Tamas Sandor Nagy, <a href="/A365236/a365236_1.png">Example for a(5)</a>.
%H Tamas Sandor Nagy, <a href="/A365236/a365236_2.png">Example for a(6)</a>.
%H Tamas Sandor Nagy, <a href="/A365236/a365236_7.png">Original upper bound examples for a(7) with 5 augmenting squares and a(8) with 6 augmenting squares</a>.
%H Tamas Sandor Nagy, <a href="/A365236/a365236_11.png">Example of a conjectured solution for a(10) with 5 augmenting squares, found by Peter Munn</a>.
%H Thomas Scheuerle, <a href="/A365236/a365236_8.png">Example for a(6) with smallest possible area</a>.
%H Thomas Scheuerle, <a href="/A365236/a365236_5.png">Example for a(7)</a>.
%H Thomas Scheuerle, <a href="/A365236/a365236_4.png">Example for a(8)</a>.
%H Thomas Scheuerle, <a href="/A365236/a365236_6.png">Example for a(9)</a>.
%H Thomas Scheuerle, <a href="/A365236/a365236_9.png">Original upper bound example for a(10) with 6 augmenting squares</a>.
%H Thomas Scheuerle, <a href="/A365236/a365236_12.png">Example for a(11) = 4 this is at the same time also a conjectured solution for a(10) = 5</a>.
%F 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).
%F a(n) > a(n - 1) - 2.
%e 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.
%e | 1^2 2^2 3^2 4^2 5^2 6^2 7^2 8^2 9^2 10^2 | Total
%e ----------------------------------------------------------------------------
%e a(1) = 0 | 1 | 1
%e a(2) = 1 | 2 1 | 3
%e a(3) = 1 | 2 1 1 | 4
%e a(4) = 3 | 2 1 2 2 | 7
%e a(5) = 2 | 2 1 1 2 1 | 7
%e a(6) = 4 | 2 1 3 2 1 1 | 10
%e a(7) = 3 | 1 1 1 3 1 2 1 | 10
%e a(8) = 3 | 3 2 1 1 1 1 1 1 | 11
%e a(9) = 4 | 2 2 2 2 1 1 1 1 1 | 13
%Y Cf. A000290, A005670, A005842, A038666, A081287.
%K nonn,more
%O 1,4
%A _Tamas Sandor Nagy_ and _Thomas Scheuerle_, Sep 25 2023
%E Edited by _Peter Munn_, Oct 04 2023
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