A386553 -id:A386553 - cp's OEIS frontend

A386554 Number of 2-colorings of an 4 X 4 X 4 grid, up to rotational symmetry, by the number of black cells.

1, 4, 98, 1744, 26691, 317728, 3125882, 25884268, 184437452, 1147524988, 6311461050, 30983159856, 136842622391, 547369124016, 1993988770170, 6646625005908, 20355293012087, 57473757334488, 150070376844312, 363328255291128, 817488598547832, 1712833203324840

Offset: 0

Marko Riedel, Jul 25 2025

This sequence is finite, having 4^3+1 terms. The cycle index Z(C) of the permutation group C of the rotations of the cube acting on the cells is given by 1/24 (a[1]^64 + 8 * a[1]^4 * a[3]^20 + 9 * a[2]^32 + 6 * a[4]^16).

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, page 35.

Sean A. Irvine, Table of n, a(n) for n = 0..64

Cf. A334616, A386553, A386555, A386556.

a(n) = [z^n] Z(C; 1+z).

A386555 Number of 2-colorings of an 5 X 5 X 5 grid, up to rotational symmetry, by the number of black cells.

Original entry on oeis.org

1, 10, 355, 13375, 404775, 9775852, 195462780, 3322598660, 49007581605, 637095486150, 7390299421967, 77262191948715, 733990748643925, 6380073195502170, 51040584989484940, 377700327297519628, 2596689746386426870, 16649363658225874915, 99896181927621097005

Offset: 0

Marko Riedel, Jul 25 2025

This sequence is finite, having 5^3+1 terms. The cycle index Z(C) of the permutation group C of the rotations of the cube acting on the cells is given by 1/24 ( a[1]^125 + 8 * a[1]^5 * a[3]^40 + 9 * a[1]^5 * a[2]^60 + 6 * a[1]^5 * a[4]^30).

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, page 35.

Sean A. Irvine, Table of n, a(n) for n = 0..125

Cf. A334616, A386553, A386554, A386556.

a(n) = [z^n] Z(C; 1+z).

A386556 Number of 2-colorings of an 6 X 6 X 6 grid / cube, up to rotational symmetry, by the number of black cells.

Original entry on oeis.org

1, 11, 1013, 69045, 3677374, 155822419, 5479820520, 164392285865, 4294750355129, 99256405950180, 2054607644379763, 38477196919023712, 657318781413490584, 10314848558604181280, 149565304110190970723, 2014146095209708440612, 25302710321203873065217, 297678944953786351579885, 3291006113657215317985320

Offset: 0

Marko Riedel, Jul 25 2025

This sequence is finite, having 6^3+1 terms. The cycle index Z(C) of the permutation group C of the rotations of the cube acting on the cells is given by 1/24 (a[1]^216 + 8 * a[1]^6 * a[3]^70 + 9 * a[2]^108 + 6 * a[4]^54).

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, page 35.

Marko Riedel, Table of n, a(n) for n = 0..216

Cf. A334616, A386553, A386554, A386555.

a(n) = [z^n] Z(C; 1+z).

cp's OEIS Frontend

A386554 Number of 2-colorings of an 4 X 4 X 4 grid, up to rotational symmetry, by the number of black cells.

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A386555 Number of 2-colorings of an 5 X 5 X 5 grid, up to rotational symmetry, by the number of black cells.

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A386556 Number of 2-colorings of an 6 X 6 X 6 grid / cube, up to rotational symmetry, by the number of black cells.

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