This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385154 #39 Aug 05 2025 10:04:27
%S A385154 10,12,21,28,28,62,42,80,57,112,114,143,90,156,191,288,184,224,252,
%T A385154 396,299,288,315,504,414,546,462,720,529,816,616,837,609,648,777,1140,
%U A385154 858,1260,874,1596,1237,1155,810,1554,1468,2064,1118,1950,1343,2080,1590,2268
%N A385154 a(n) is the least possible difference between the largest and smallest volumes of distinct five-cuboid combinations filling an n X n X n cube.
%C A385154 Developed as the three-dimensional extension of the Mondrian Art Problem.
%C A385154 Alternatively, a(n) is the optimal solution when an n X n X n cube is partitioning into five cuboids of different dimensions.
%C A385154 Let elements of the unordered integer triplet (x,y,z) be the dimensions of a cuboid in a set of five cuboids and volume V(x,y,z) = x*y*z; cuboids have five values for each set of five triplets S, produced by the union of A(n), B(n), C(n), where A(n), B(n), and C(n) are sequences of sets as introduced in A384479.
%C A385154 Define min(S) = min{V(x,y,z):(x,y,z) in S} and max(S) = max{V(x,y,z):(x,y,z) in S}, then defect(S) = max(S) - min(S).
%C A385154 a(n) is the smallest possible value of defect(S) where S runs over all possible ways of partitioning the n X n X n cube into five cuboids of different dimensions.
%H A385154 Janaka Rodrigo, <a href="/A385154/a385154.txt">Python Code for Minimum defects of A(n)</a>
%H A385154 Janaka Rodrigo, <a href="/A385154/a385154_1.txt">Python Code for Minimum defects of B(n)</a>
%H A385154 Janaka Rodrigo, <a href="/A385154/a385154_2.txt">Python Code for Minimum defects of C(n) </a>
%e A385154 4 X 4 X 4 cube has 31 different ways of partitioning into five distinct cuboids and only two sets producing the minimum defects as calculated below:
%e A385154 {(3,2,1), (3,3,2), (4,1,2), (4,2,2),(4,4,1)} has minimum defect = max(6,18,8,16,16) - min(6,18,8,16,16) = 18 - 6 = 12.
%e A385154 {(4,4,1), (4,3,1), (2,3,3), (2,3,2), (2,3,1)} has minimum defect = max(16,12,18,12,6) - min(16,12,18,12,6) = 18 - 6 = 12.
%e A385154 Therefore a(4) = 12.
%Y A385154 Cf. A276523, A384479.
%K A385154 nonn,hard
%O A385154 3,1
%A A385154 _Janaka Rodrigo_, Jun 19 2025
%E A385154 a(11)-a(20) from _Sean A. Irvine_, Jul 26 2025
%E A385154 a(21)-a(54) from _Jinyuan Wang_, Aug 04 2025