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.

1, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 21, 30, 29, 20, 25, 30, 12, 19, 24, 17, 13, 13, 18, 14, 19, 14, 15, 15, 15, 20, 15, 20, 16, 22, 16, 16, 17, 21, 22, 15, 13, 16, 18, 14, 14, 14, 17, 15, 11, 10, 12, 13, 4, 11, 8, 9, 7, 11, 4, 9, 8, 8, 8, 6, 8

Offset: 1

Author

Vitor Pimenta dos Reis Arruda, May 15 2020

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• Vitor Pimenta dos Reis Arruda, Table of n, a(n) for n = 1..101
• I. Gambini, A method for cutting squares into distinct squares, Discrete Applied Mathematics, 98 (1999), 65-80.
• Vitor Pimenta dos Reis Arruda, Non trivial decompositions until a(101)
• Vitor Pimenta dos Reis Arruda, Luiz Gustavo Bizarro Mirisola, and Nei Yoshihiro Soma, Almost squaring the square: optimal packings for non-decomposable squares, Pesqui. Oper. (2022) Vol. 42.
• Giovanni Resta, Illustration of terms a(15)-a(31)
• Wikipedia, Squaring the square

Crossrefs

Cf. A006983, A002962, A181735, A002839, A228953.

Extensions

Terms a(17)-a(31) from Giovanni Resta, May 15 2020