A193416 -id:A193416 - cp's OEIS frontend

A007818 Maximal number of bonds joining n nodes in simple cubic lattice.

0, 1, 2, 4, 5, 7, 9, 12, 13, 15, 17, 20, 21, 23, 25, 28, 30, 33, 34, 36, 38, 41, 43, 46, 48, 51, 54, 55, 57, 59, 62, 64, 67, 69, 72, 75, 76, 78, 80, 83, 85, 88, 90, 93, 96, 98, 101, 104, 105, 107, 109, 112, 114, 117, 119, 122, 125, 127, 130, 133, 135, 138, 141

Offset: 1

D. Heuer (heuer(AT)isnd23.in2p3.fr)

nonn

a(n) is also the maximal number of kisses between n cubes to form a polycube. The surface area of such polycubes are A193416. - Mohammed Yaseen, Aug 08 2021

Martin Y. Veillette, Table of n, a(n) for n = 1..500
G. Agnarsson, On the number of hypercubic bipartitions of an integer, arXiv preprint arXiv:1106.4997 [math.CO], 2011.
G. Agnarsson, Induced subgraphs of hypercubes, arXiv preprint arXiv:1112.3015 [math.CO], 2011.
G. Agnarsson and K. Lauria, Extremal subgraphs of the d-dimensional grid graph, arXiv preprint arXiv:1302.6517 [math.CO], 2013.

Cf. A193416.

qmax = 2000; sequence =
FoldList[Plus, 0, q = Table[3, {qmax}];
  q[[Flatten[
     Table[Table[{j^3 + i (i - 1), j^3 + i^2, j^2 (j + 1) + i (i + 1),
         j^2 (j + 1) + i^2, (j + 1)^2 j + i (i + 1), (j + 1)^2 j +
         i^2}, {i, 1, j}], {j, 0, (qmax)^(1/3) - 1}]]]]--;
  q[[Flatten[
     Table[{j^3, j^2 (j + 1), (j + 1)^2 j}, {j,
       1, (qmax)^(1/3) - 1}]]]]--; q] (* Martin Y. Veillette, Jul 19 2011 *)

a(n) = 3*n - A193416(n)/2. - Mohammed Yaseen, Aug 08 2021

A346958 a(n) is the minimal number of cubes required to make a void of volume n.

Original entry on oeis.org

6, 10, 13, 15, 17, 18, 18, 21, 23, 25, 26, 26

Offset: 1

Mohammed Yaseen, Aug 08 2021

Following is an illustration of the first few voids in the form of polycubes (where an o represents a continuation upwards and an x represents a continuation downwards) each of which can be made by concealing it with a(n) cubes.
                          .---.       .---.
                          |   |       |   |
  .---.   .---.---.   .---.---.   .---.---.
  |   |   |   |   |   |   |   |   |   | o |
  .---.   .---.---.   .---.---.   .---.---.
   n=1       n=2         n=3         n=4
      .---.       .---.           .---.
      |   |       |   |           |   |
  .---.---.   .---.---.---.   .---.---.---.
  |   | o |   |   | o |   |   |   | ox|   |
  .---.---.   .---.---.---.   .---.---.---.
      |   |       |   |           |   |
      .---.       .---.           .---.
     n=5           n=6             n=7
Equivalently, the minimum perimeter size of any polycube of size n. - Sean A. Irvine, Aug 23 2021
Conjecture: When n is in A001845 the void is an octahedral crystal ball of volume n = A001845(m), which is concealed by a(n) = A005899(m+1) cubes. So a(A001845(m)) = A005899(m+1), m>=0. For example, a(1)=6 and a(7)=18. - Mohammed Yaseen, Sep 15 2022

			A cube-shaped void can be made by concealing it with 6 cubes, which is the minimal number to do so. So a(1)=6.
A dicube-shaped void can be made by concealing it with 10 cubes, which is the minimal number to do so. So a(2)=10.

Sean A. Irvine, Java program (github)

Cf. A261491 (2D analog).

Cf. A000162, A003211, A193416.

a(n) < A193416(n) for n>2.

a(8)-a(12) from Sean A. Irvine, Aug 23 2021

A363609 Minimum sum of the visible pips on a polycube made from n dice.

Original entry on oeis.org

21, 30, 40, 40, 51, 52, 54, 48, 60, 62, 65, 60, 72, 74, 77, 72, 78, 74, 86, 84, 91, 88, 92, 88, 95, 93, 90, 102, 105, 102, 106, 104, 107, 110, 109, 106, 118, 120, 121, 120, 125, 123, 126, 125, 122, 128, 127, 124, 136, 138, 139, 140, 141, 138, 145, 144, 143

Offset: 1

Matt Donahoe, Jun 11 2023

nonn

This sequence is calculated using standard six-sided dice of the same chirality. Opposite sides sum to seven.

			For n = 2, two dice are conjoined to hide both their 6-pip faces, so a(2) = 30.
For n = 4, four dice are arranged in a 2 X 2 square such that no 5-pip or 6-pip faces are visible. When the dice can form a cube, such as n = 8, only 1-, 2- and 3-pip faces will be visible.

Matt Donahoe, Python program
Wikipedia, Dice

Conceptually similar to A193416.

Python
```
# see linked program
```

Conjecture: a(k^3) = 6*(k+2)*k for k > 1.
a(i*j*k) <= 48 + 2*((i-2)*(j-2) + (i-2)*(k-2) + (j-2)*(k-2)) + 12*(i+j+k-6), for i, j, k > 1. - Michael S. Branicky, Jun 15 2023
A193416(n) <= lb(n) <= a(n) <= ub(n) <= 6*A193416(n), where:
  lb(n) = Sum_{i=1..A193416(n)} S(i, n),
  ub(n) = Sum_{i=1..A193416(n)} S(6*n+1-i, n), and
  S(i, j) = 1 + floor((i-1)/j). - Michael S. Branicky, Jun 11 2023

a(22)-a(24) corrected by Michael S. Branicky, Jun 18 2023

A386400 Minimum volume of n polycubes such that each polycube is adjacent to every other polycube.

Original entry on oeis.org

1, 2, 4, 7, 13, 20, 28, 36

Offset: 1

David Hathaway, Jul 20 2025

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).
Volume in measured by the total number of cells.
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

			a(n) can be seen by the following diagrams, where each cell in the i-th polycube is labeled by [i].
a(1)=1:
 [1]
a(2)=2:
 [1][2]
a(3)=4:
 [1][3]
 [2][3]
a(4)=7:
 Layer 1  Layer 2
 [1][3]   [4][4]
 [2][3]   [2]
a(5)=13:
 Layer 1    Layer 2
 [5][1][3]  [4][4][4]
 [5][2][3]     [2]
 [5][3][3]

Wikipedia, Hadwiger number.

Cf. A193416.

a(n) <= n^2-n-6, for n>4
This upper bound comes from the following construction which holds for all n>4:
Layer 1                       Layer 2
[1][1]... [1]  [1]  [1] [1]                           [n]
[1][4]...[n-3][n-2][n-1]      [2][2]... [2]  [2]  [2] [n]
[3][4]...[n-3][n-2][n-1]      [3][3]... [3]  [3]  [3] [n]
   [4]...[n-3][n-2][n-1]         [4]... [4]  [4]  [4] [n]
      ...  :    :    :              ...  :    :    :   :
         [n-3][n-2][n-1]               [n-3][n-3][n-3][n]
              [n-2][n-1]                    [n-2] [n] [n]

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A007818 Maximal number of bonds joining n nodes in simple cubic lattice.

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A346958 a(n) is the minimal number of cubes required to make a void of volume n.

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A363609 Minimum sum of the visible pips on a polycube made from n dice.

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Python

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A386400 Minimum volume of n polycubes such that each polycube is adjacent to every other polycube.

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