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 A386400 #25 Aug 15 2025 15:46:54 %S A386400 1,2,4,7,13,20,28,36 %N A386400 Minimum volume of n polycubes such that each polycube is adjacent to every other polycube. %C A386400 Two polycubes are adjacent if there is at least one cell in the first polycube and at least one cell in the second polycube that are aligned such that they can be joined face to face (i.e., a third polycube can be made with those two cells). %C A386400 Volume in measured by the total number of cells. %C A386400 Equivalently, a(n) is the least number of vertices in a subgraph of the 3-dimensional grid that has Hadwiger number n. - _Pontus von Brömssen_, Jul 31 2025 %H A386400 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hadwiger_number">Hadwiger number</a>. %F A386400 a(n) <= n^2-n-6, for n>4 %F A386400 This upper bound comes from the following construction which holds for all n>4: %F A386400 Layer 1 Layer 2 %F A386400 [1][1]... [1] [1] [1] [1] [n] %F A386400 [1][4]...[n-3][n-2][n-1] [2][2]... [2] [2] [2] [n] %F A386400 [3][4]...[n-3][n-2][n-1] [3][3]... [3] [3] [3] [n] %F A386400 [4]...[n-3][n-2][n-1] [4]... [4] [4] [4] [n] %F A386400 ... : : : ... : : : : %F A386400 [n-3][n-2][n-1] [n-3][n-3][n-3][n] %F A386400 [n-2][n-1] [n-2] [n] [n] %e A386400 a(n) can be seen by the following diagrams, where each cell in the i-th polycube is labeled by [i]. %e A386400 a(1)=1: %e A386400 [1] %e A386400 a(2)=2: %e A386400 [1][2] %e A386400 a(3)=4: %e A386400 [1][3] %e A386400 [2][3] %e A386400 a(4)=7: %e A386400 Layer 1 Layer 2 %e A386400 [1][3] [4][4] %e A386400 [2][3] [2] %e A386400 a(5)=13: %e A386400 Layer 1 Layer 2 %e A386400 [5][1][3] [4][4][4] %e A386400 [5][2][3] [2] %e A386400 [5][3][3] %Y A386400 Cf. A193416. %K A386400 nonn,more %O A386400 1,2 %A A386400 _David Hathaway_, Jul 20 2025