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A334905 a(n) is the minimum remaining space when a square n X n is tiled with smaller squares with distinct integer sides parallel to the n X n square.

%I A334905 #45 Nov 29 2022 01:22:21
%S A334905 1,3,4,6,8,10,12,14,16,18,20,22,24,26,21,30,29,20,25,30,12,19,24,17,
%T A334905 13,13,18,14,19,14,15,15,15,20,15,20,16,22,16,16,17,21,22,15,13,16,18,
%U A334905 14,14,14,17,15,11,10,12,13,4,11,8,9,7,11,4,9,8,8,8,6,8
%N A334905 a(n) is the minimum remaining space when a square n X n is tiled with smaller squares with distinct integer sides parallel to the n X n square.
%C A334905 See (Gambini, 1999) for a way to construct the sequence. Actually, one would have to extend Gambini's idea by putting extra 1-sided squares in the list of "usable squares" to allow finding nonzero-waste packings.
%H A334905 Vitor Pimenta dos Reis Arruda, <a href="/A334905/b334905.txt">Table of n, a(n) for n = 1..101</a>
%H A334905 I. Gambini, <a href="https://doi.org/10.1016/S0166-218X(99)00158-4">A method for cutting squares into distinct squares</a>, Discrete Applied Mathematics, 98 (1999), 65-80.
%H A334905 Vitor Pimenta dos Reis Arruda, <a href="/A334905/a334905_3.pdf">Non trivial decompositions until a(101)</a>
%H A334905 Vitor Pimenta dos Reis Arruda, Luiz Gustavo Bizarro Mirisola, and Nei Yoshihiro Soma, <a href="https://doi.org/10.1590/0101-7438.2022.042.00262876">Almost squaring the square: optimal packings for non-decomposable squares</a>, Pesqui. Oper. (2022) Vol. 42.
%H A334905 Giovanni Resta, <a href="/A334905/a334905.pdf">Illustration of terms a(15)-a(31)</a>
%H A334905 Wikipedia, <a href="https://www.wikipedia.org/wiki/Squaring_the_square">Squaring the square</a>
%e A334905 For n=5, squares of sides {1, 4} can be packed inside the container, leading to uncovered area a(5) = 5*5 - (4*4 + 1*1) = 8. The other maximal packable set is composed of the squares sided {1,2,3}, which would lead to uncovered area greater than 8.
%Y A334905 Cf. A006983, A002962, A181735, A002839, A228953.
%K A334905 nonn
%O A334905 1,2
%A A334905 _Vitor Pimenta dos Reis Arruda_, May 15 2020
%E A334905 Terms a(17)-a(31) from _Giovanni Resta_, May 15 2020