A000162 -id:A000162 - cp's OEIS frontend

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 12

Offset: 1

John Mason, Oct 17 2024

See link "Counting free polycubes" for explanation of notation.

John Mason, Counting free polycubes

Cf. A000162, A038119.

A038386 Number of "connected animals" formed from n 4-gon or 6-gon connected truncated octahedra in the b.c.c. lattice, allowing only translation of the lattice.

Original entry on oeis.org

1, 7, 67, 734, 8716, 109070, 1416284, 18903142, 257718235, 3573580367

Offset: 1

Torsten Sillke (TORSTEN.SILLKE(AT)LHSYSTEMS.COM)

S. T. Coffin, Puzzling World of Polyhedral Dissections, Oxford Univ. Press, 1991.
T. Sillke, Notations for polyspheres
Index entries for sequences related to b.c.c. lattice

Cf. A000162, A038119, A038168-A038174, A038180, A038181.

More terms from Achim Flammenkamp

Name corrected and a(9)-a(10) from Aaron N. Siegel, May 31 2022

A122675 Number of "fragments" with n nodes generated from the simple cubic lattice.

Original entry on oeis.org

1, 1, 2, 9, 29, 165, 962, 6423

Offset: 1

N. J. A. Sloane, Sep 23 2006

The "fragments" are generated as follows. For each of the polycubes with n cells, counted by A000162(n) or b(n) = A038119(n), consider two possible ways to inscribe a tetrahedron into each cell so that the tetrahedra in any two neighboring cells share an edge. The centers of the cells correspond to cations in the antifluorite structure, while the vertices of the tetrahedra correspond to anions. a(n) is the number of resulting tetrahedral clusters; enantiomorphic pairs are counted as one. Thus b(n) <= a(n) <= 2*b(n). - Andrey Zabolotskiy, Mar 04 2023

J. R. Long and R. H. Holm, Enumeration and Structural Classification of Clusters Derived from Parent Solids: Metal-Chalcogenide Clusters Composed of Edge-Sharing Tetrahedra, J. Amer. Chem. Soc., 116 (1994), 9987-10002.
Wikipedia, Fluorite structure

Cf. A000162, A038119, A122673, A122674.

A272385 Number of polycubes with n cells, allowing vertex connections and edge connections as well as face connections, distinguishing mirror images.

Original entry on oeis.org

1, 3, 16, 246, 4866, 115520, 2873712, 73884172, 1941259395

Offset: 1

George Sicherman, Apr 28 2016

			For n=2 there are three possibilities: the two cubes share a face, or only an edge, or only a vertex.
For n=3, there are 14 tricubes allowing vertex, edge, and face connections. Two of them have distinct mirror images, so a(3) = 16.

Cf. A272368 (identifying mirror images), A270862 (no vertex connections), A000162 (face connections only).

a(8) and a(9) from Joerg Arndt, Dec 13 2023

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

cp's OEIS Frontend

A376988 Number of polycubes of size n and symmetry class EF.

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A376989 Number of polycubes of size n and symmetry class EFF.

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A376990 Number of polycubes of size n and symmetry class CF.

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A376991 Number of polycubes of size n and symmetry class BBC.

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A377127 Number of polycubes of size n and symmetry class AB.

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A377131 Number of polycubes of size n and symmetry class R.

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A038386 Number of "connected animals" formed from n 4-gon or 6-gon connected truncated octahedra in the b.c.c. lattice, allowing only translation of the lattice.

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A122675 Number of "fragments" with n nodes generated from the simple cubic lattice.

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A272385 Number of polycubes with n cells, allowing vertex connections and edge connections as well as face connections, distinguishing mirror images.

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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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