A331351 -id:A331351 - cp's OEIS frontend

A060530 Number of inequivalent ways to color edges of a cube using at most n colors.

0, 1, 218, 22815, 703760, 10194250, 90775566, 576941778, 2863870080, 11769161895, 41669295250, 130772947481, 371513523888, 970769847320, 2362273657030, 5406141568500, 11728193258496, 24276032182173, 48201464902410, 92221684354915

Offset: 0

N. J. A. Sloane, Apr 11 2001

Here inequivalent means under the action of the rotation group of the cube, of order 24, which in its action on the edges has cycle index (x1^12 + 3*x2^6 + 6*x4^3 + 6*x1^2*x2^5 + 8*x3^4)/24.
Also, number of inequivalent colorings of the edges of a regular octahedron using at most n colors. - José H. Nieto S., Jan 19 2012
From Robert A. Russell, Oct 08 2020: (Start)
Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbols for the regular octahedron and cube are {3,4} and {4,3} respectively. They are mutually dual.
There are 24 elements in the rotation group of the regular octahedron/cube. They divide into five conjugacy classes. The first formula is obtained by averaging the edge cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Conjugacy Class    Count    Even Cycle Indices
  Identity              1     x_1^12
  Vertex rotation       8     x_3^4
  Edge rotation         6     x_1^2x_2^5
  Small face rotation   6     x_4^3
  Large face rotation   3     x_2^6 (End)
Also, number of ways of coloring the vertices of the truncated tetrahedron or faces of the triakis tetrahedron up to rotation and reflection. - Peter Kagey, Nov 27 2024

N. G. De Bruijn, Polya's theory of counting, in E. F. Beckenbach, ed., Applied Combinatorial Mathematics, Wiley, 1964, pp. 144-184 (see p. 147).

Harry J. Smith, Table of n, a(n) for n=0..200
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A199406 (unoriented), A337406 (chiral), A331351 (achiral).
Other elements: A000543 (cube vertices, octahedron faces), A047780 (cube faces, octahedron vertices).
Cf. A046023 (tetrahedron), A282670 (dodecahedron/icosahedron).
Row 3 of A337407 (orthotope edges, orthoplex ridges) and A337411 (orthoplex edges, orthotope ridges).

Table[(n^12+6n^7+3n^6+8n^4+6n^3)/24,{n,0,20}] (* Harvey P. Dale, Feb 13 2013 *)

{ for (n=0, 200, write("b060530.txt", n, " ", (n^12 + 6*n^7 + 3*n^6 + 8*n^4 + 6*n^3)/24); ) } \\ Harry J. Smith, Jul 06 2009

a(n) = (n^12 + 6*n^7 + 3*n^6 + 8*n^4 + 6*n^3)/24. (Replace all x_i's in the cycle index by n.)
G.f.: -x*(150*x^10 +19758*x^9 +425032*x^8 +2763481*x^7 +6769435*x^6 +6773089*x^5 +2763307*x^4 +423883*x^3 +20059*x^2 +205*x +1)/(x -1)^13. - Colin Barker, Aug 13 2012
From Robert A. Russell, Oct 08 2020: (Start)
a(n) = 1*C(n,1) + 216*C(n,2) + 22164*C(n,3) + 613804*C(n,4) + 6901425*C(n,5) + 39713430*C(n,6) + 131754420*C(n,7) + 267165360*C(n,8) + 336798000*C(n,9) + 257796000*C(n,10) + 109771200*C(n,11) + 19958400*C(n,12), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
a(n) = A199406(n) + A337406(n) = 2*A199406(n) - A331351(n) = 2*A337406(n) + A331351(n). (End)

Entry revised by N. J. A. Sloane, Jan 03 2005

A199406 The number of inequivalent ways to color the edges of a cube using at most n colors.

Original entry on oeis.org

1, 144, 12111, 358120, 5131650, 45528756, 288936634, 1433251296, 5887880415, 20842168600, 65402344161, 185788177224, 485443851256, 1181242399260, 2703252560100, 5864398969216, 12138503871789, 24101498435616, 46112016365155, 85335258695400, 153249227870046

Offset: 1

Geoffrey Critzer, Nov 05 2011

Two edge colorings are equivalent if one is the mirror image of the other or the cube can be picked up and rotated in any manner to obtain the other.
The group here has order 48 (compare A060530). - N. J. A. Sloane, Aug 14 2012
Also the number of unoriented colorings of the 12 edges of a regular octahedron with n or fewer colors. The Schläfli symbols of the cube and octahedron are {4,3} and {3,4} respectively. They are mutually dual. For an unoriented coloring, chiral pairs are counted as one. - Robert A. Russell, Oct 17 2020

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A060530 (oriented), A337406 (chiral), A331351 (achiral), A128766 (cube vertices, octahedron faces), A198833 (cube faces, octahedron vertices), A063842(n-1) (tetrahedron), A337963 (dodecahedron, icosahedron).

Row 3 of A337408 (orthotope edges, orthoplex ridges) and A337412 (orthoplex edges, orthotope ridges).

Table[CycleIndex[KSubsetGroup[Automorphisms[CubicalGraph], Edges[CubicalGraph]],s] /. Table[s[i]->n, {i,1,6}], {n,1,15}]
Table[(8n^2+12n^3+8n^4+4n^6+12n^7+3n^8+n^12)/48, {n,20}] (* Robert A. Russell, Oct 17 2020 *)

a(n) = n^12/48 + n^8/16 + n^7/4 + n^6/12 + n^4/6 + n^3/4 + n^2/6.
Cycle index = 1/48(s_1^12+3s_1^4s_2^4+12s_1^2s_2^5+4s_2^6+8s_3^4+12s_4^3+8s_6^2).
G.f.: -x*(76*x^10 +10016*x^9 +212772*x^8 +1380453*x^7 +3384939*x^6 +3388593*x^5 +1380279*x^4 +211623*x^3 +10317*x^2 +131*x +1)/(x -1)^13. [Colin Barker, Aug 13 2012]
From Robert A. Russell, Oct 17 2020: (Start)
a(n) = A060530(n) - A337406(n) = (A060530(n) + A331351(n)) / 2 = A337406(n) + A331351(n).
a(n) = 1*C(n,1) + 142*C(n,2) + 11682*C(n,3) + 310536*C(n,4) + 3460725*C(n,5) + 19870590*C(n,6) + 65886660*C(n,7) + 133585200*C(n,8) + 168399000*C(n,9) + 128898000*C(n,10) + 54885600*C(n,11) + 9979200*C(n,12), where the coefficient of C(n,k) is the number of unoriented colorings using exactly k colors. (End)

A337414 Array read by descending antidiagonals: T(n,k) is the number of achiral colorings of the edges of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.

Original entry on oeis.org

1, 2, 1, 3, 6, 1, 4, 18, 70, 1, 5, 40, 1407, 8200, 1, 6, 75, 12480, 9080559, 12804908, 1, 7, 126, 69050, 1503323520, 4906480368591, 304899216832, 1, 8, 196, 281946, 81461669375, 48226825456539776, 187380251418565888983, 103685962258536432, 1

Offset: 1

Robert A. Russell, Aug 26 2020

An achiral arrangement is identical to its reflection. For n=1, the figure is a line segment with one edge. For n=2, the figure is a square with 4 edges. For n=3, the figure is an octahedron with 12 edges. The number of edges is 2n*(n-1) for n>1.

Also the number of achiral colorings of the regular (n-2)-dimensional orthotopes (hypercubes) in a regular n-dimensional orthotope.

			Table begins with T(1,1):
1  2    3     4     5      6      7       8       9       10 ...
1  6   18    40    75    126    196     288     405      550 ...
1 70 1407 12480 69050 281946 931490 2632512 6598935 15041950 ...
For T(2,2)=6, the arrangements are AAAA, AAAB, AABB, ABAB, ABBB, and BBBB.

K. Balasubramanian, Computational enumeration of colorings of hyperplanes of hypercubes for all irreducible representations and applications, J. Math. Sci. & Mod. 1 (2018), 158-180.

Cf. A337411 (oriented), A337412 (unoriented), A337413 (chiral).
Rows 1-4 are A000027, A002411, A331351, A331357.
Cf. A327086 (simplex edges), A337410 (orthotope edges), A325007 (orthoplex vertices).

m=1; (* dimension of color element, here an edge *)
Fi1[p1_] := Module[{g, h}, Coefficient[Product[g = GCD[k1, p1]; h = GCD[2 k1, p1]; (1 + 2 x^(k1/g))^(r1[[k1]] g) If[Divisible[k1, h], 1, (1+2x^(2 k1/h))^(r2[[k1]] h/2)], {k1, Flatten[Position[cs, n1_ /; n1 > 0]]}], x, m+1]];
FiSum[] := (Do[Fi2[k2] = Fi1[k2], {k2, Divisors[per]}];DivisorSum[per, DivisorSum[d1 = #, MoebiusMu[d1/#] Fi2[#] &]/# &]);
CCPol[r_List] := (r1 = r; r2 = cs - r1; If[EvenQ[Sum[If[EvenQ[j3], r1[[j3]], r2[[j3]]], {j3,n}]],0,(per = LCM @@ Table[If[cs[[j2]] == r1[[j2]], If[0 == cs[[j2]],1,j2], 2j2], {j2,n}]; Times @@ Binomial[cs, r1] 2^(n-Total[cs]) b^FiSum[])]);
PartPol[p_List] := (cs = Count[p, #]&/@ Range[n]; Total[CCPol[#]&/@ Tuples[Range[0,cs]]]);
pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #]&/@ mb; n!/(Times@@(ci!) Times@@(mb^ci))] (*partition count*)
row[m]=b;
row[n_Integer] := row[n] = Factor[(Total[(PartPol[#] pc[#])&/@ IntegerPartitions[n]])/(n! 2^(n-1))]
array[n_, k_] := row[n] /. b -> k
Table[array[n,d+m-n], {d,8}, {n,m,d+m-1}] // Flatten

The algorithm used in the Mathematica program below assigns each permutation of the axes to a partition of n and then considers separate conjugacy classes for axis reversals. It uses the formulas in Balasubramanian's paper. If the value of m is increased, one can enumerate colorings of higher-dimensional elements beginning with T(m,1).

T(n,k) = 2*A337412(n,k) - A337411(n,k) = A337411(n,k) - 2*A337413(n,k) = A337412(n,k) - A337413(n,k).

A337897 Number of achiral colorings of the 8 triangular faces of a regular octahedron or the 8 vertices of a cube using n or fewer colors.

Original entry on oeis.org

1, 21, 201, 1076, 4025, 11901, 29841, 66256, 134001, 251725, 445401, 750036, 1211561, 1888901, 2856225, 4205376, 6048481, 8520741, 11783401, 16026900, 21474201, 28384301, 37055921, 47831376, 61100625, 77305501, 96944121

Offset: 1

Robert A. Russell, Sep 28 2020

nonn

An achiral coloring is identical to its reflection. The Schläfli symbols for the cube and regular octahedron are {4,3} and {3,4} respectively. They are mutually dual.
There are 24 elements in the automorphism group of the regular octahedron/cube that are not in the rotation group. They divide into five conjugacy classes. The first formula is obtained by averaging the cube vertex (octahedron face) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Conjugacy Class     Count    Odd Cycle Indices
  Inversion              1     x_2^4
  Vertex rotation*       8     x_2^1x_6^1      Asterisk indicates that the
  Edge rotation*         6     x_1^4x_2^2      operation is followed by an
  Small face rotation*   3     x_4^2           inversion.
  Large face rotation*   6     x_2^4

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A000543 (oriented), A128766 (unoriented), A337896 (chiral).
Other elements: A331351 (edges), A337898 (cube faces, octahedron vertices).
Other polyhedra: A006003 (tetrahedron), A337962  (dodecahedron faces, icosahedron vertices), A337960 (icosahedron faces, dodecahedron vertices).
Row 3 of A337894 (orthoplex faces, orthotope peaks) and A325015 (orthotope vertices, orthoplex facets).

Mathematica
```
Table[n^2(7+2n^2+3n^4)/12, {n,30}]
```

a(n) = n^2 * (7 + 2*n^2 + 3*n^4) / 12.
a(n) = 1*C(n,1) + 19*C(n,2) + 141*C(n,3) + 394*C(n,4) + 450*C(n,5) + 180*C(n,6), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n) = 2*A128766(n) - A000543(n) = A000543(n) - 2*A337896(n) = A128766(n) - A337896(n).
G.f.: x * (1+x) * (1 + 13*x + 62*x^2 + 13*x^3 + x^4) / (1-x)^7.

A337406 Number of chiral pairs of colorings of the edges of a cube (or regular octahedron) using n or fewer colors.

Original entry on oeis.org

0, 74, 10704, 345640, 5062600, 45246810, 288005144, 1430618784, 5881281480, 20827126650, 65370603320, 185725346664, 485325996064, 1181031257770, 2702889008400, 5863794289280, 12137528310384, 24099966466794

Offset: 1

Robert A. Russell, Aug 26 2020

Each member of a chiral pair is a reflection, but not a rotation, of the other. Both the cube and the octahedron have 12 edges.

Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Cf. A060530 (oriented), A199406 (unoriented), A331351 (achiral).

Row 3 of A337409 (orthotope edge colorings) and A337413 (orthoplex edge colorings).

Table[(n-1)n^2(n+1)(n^8+n^6-2n^4+8)/48, {n,20}]
LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{0,74,10704,345640,5062600,45246810,288005144,1430618784,5881281480,20827126650,65370603320,185725346664,485325996064},20] (* Harvey P. Dale, Jul 11 2025 *)

a(n) = (n-1) * n^2 * (n+1) * (n^8 + n^6 - 2n^4 + 8) / 48.
a(n) = 74*C(n,2) + 10482*C(n,3) + 303268*C(n,4) + 3440700*C(n,5) + 19842840*C(n,6) + 65867760*C(n,7) + 133580160*C(n,8) + 168399000*C(n,9) + 128898000*C(n,10) + 54885600*C(n,11) + 9979200*C(n,12), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = (A060530(n) - A331351(n)) / 2 = A060530(n) - A199406(n) = A199406(n) - A331351(n).
G.f.: 2 * (37*x^2 + 4871*x^3 + 106130*x^4 + 691514*x^5 + 1692248*x^6 + 1692248*x^7 + 691514*x^8 + 106130*x^9 + 4871*x^10 + 37*x^11) / (1-x)^13.

A337410 Array read by descending antidiagonals: T(n,k) is the number of achiral colorings of the edges of a regular n-dimensional orthotope (hypercube) using k or fewer colors.

Original entry on oeis.org

1, 2, 1, 3, 6, 1, 4, 18, 70, 1, 5, 40, 1407, 93024, 1, 6, 75, 12480, 294157089, 47823602694208, 1, 7, 126, 69050, 91983927296, 67514530382043163023924, 443077371786837979607993095063601152, 1

Offset: 1

Robert A. Russell, Aug 26 2020

An achiral arrangement is identical to its reflection. For n=1, the figure is a line segment with one edge. For n=2, the figure is a square with 4 edges. For n=3, the figure is a cube with 12 edges. The number of edges is n*2^(n-1).

Also the number of achiral colorings of the regular (n-2)-dimensional simplexes in a regular n-dimensional orthoplex.

			Table begins with T(1,1):
1  2    3     4     5      6      7       8       9       10 ...
1  6   18    40    75    126    196     288     405      550 ...
1 70 1407 12480 69050 281946 931490 2632512 6598935 15041950 ...
For T(2,2)=6, the arrangements are AAAA, AAAB, AABB, ABAB, ABBB, and BBBB.

K. Balasubramanian, Computational enumeration of colorings of hyperplanes of hypercubes for all irreducible representations and applications, J. Math. Sci. & Mod. 1 (2018), 158-180.

Cf. A337407 (oriented), A337408 (unoriented), A337409 (chiral).
Rows 1-4 are A000027, A002411, A331351, A331361.
Cf. A327086 (simplex edges), A337414 (orthoplex edges), A325015 (orthotope vertices).

m=1; (* dimension of color element, here an edge *)
Fi1[p1_] := Module[{g, h}, Coefficient[Product[g = GCD[k1, p1]; h = GCD[2 k1, p1]; (1 + 2 x^(k1/g))^(r1[[k1]] g) If[Divisible[k1, h], 1, (1+2x^(2 k1/h))^(r2[[k1]] h/2)], {k1, Flatten[Position[cs, n1_ /; n1 > 0]]}], x, n - m]];
FiSum[] := (Do[Fi2[k2] = Fi1[k2], {k2, Divisors[per]}];DivisorSum[per, DivisorSum[d1 = #, MoebiusMu[d1/#] Fi2[#] &]/# &]);
CCPol[r_List] := (r1 = r; r2 = cs - r1; If[EvenQ[Sum[If[EvenQ[j3], r1[[j3]], r2[[j3]]], {j3,n}]],0,(per = LCM @@ Table[If[cs[[j2]] == r1[[j2]], If[0 == cs[[j2]],1,j2], 2j2], {j2,n}]; Times @@ Binomial[cs, r1] 2^(n-Total[cs]) b^FiSum[])]);
PartPol[p_List] := (cs = Count[p, #]&/@ Range[n]; Total[CCPol[#]&/@ Tuples[Range[0,cs]]]);
pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #]&/@ mb; n!/(Times@@(ci!) Times@@(mb^ci))] (*partition count*)
row[n_Integer] := row[n] = Factor[(Total[(PartPol[#] pc[#])&/@ IntegerPartitions[n]])/(n! 2^(n-1))]
array[n_, k_] := row[n] /. b -> k
Table[array[n,d+m-n], {d,7}, {n,m,d+m-1}] // Flatten

The algorithm used in the Mathematica program below assigns each permutation of the axes to a partition of n and then considers separate conjugacy classes for axis reversals. It uses the formulas in Balasubramanian's paper. If the value of m is increased, one can enumerate colorings of higher-dimensional elements beginning with T(m,1).

T(n,k) = 2*A337408(n,k) - A337407(n,k) = A337407(n,k) - 2*A337409(n,k) = A337408(n,k) - A337409(n,k).

A337898 Number of achiral colorings of the 6 square faces of a cube or the 6 vertices of a regular octahedron using n or fewer colors.

Original entry on oeis.org

1, 10, 55, 200, 560, 1316, 2730, 5160, 9075, 15070, 23881, 36400, 53690, 77000, 107780, 147696, 198645, 262770, 342475, 440440, 559636, 703340, 875150, 1079000, 1319175, 1600326, 1927485, 2306080, 2741950, 3241360

Offset: 1

Robert A. Russell, Sep 28 2020

An achiral coloring is identical to its reflection. The Schläfli symbols for the cube and regular octahedron are {4,3} and {3,4} respectively. They are mutually dual.
There are 24 elements in the automorphism group of the regular octahedron/cube that are not in the rotation group. They divide into five conjugacy classes. The first formula is obtained by averaging the cube face (octahedron vertex) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Conjugacy Class     Count    Odd Cycle Indices
  Inversion              1     x_2^3
  Vertex rotation*       8     x_6^1           Asterisk indicates that the
  Edge rotation*         6     x_1^2x_2^2      operation is followed by an
  Small face rotation*   6     x_2^1x_4^1      inversion.
  Large face rotation*   3     x_1^4x_2^1

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A047780 (oriented), A198833 (unoriented), A093566(n+1) (chiral).
Other elements: A331351 (edges), A337897 (cube vertices/octahedron faces).
Other polyhedra: A006003 (simplex), A337962 (dodecahedron faces, icosahedron vertices), A337960 (icosahedron faces, dodecahedron vertices).
Row 3 of A325007 (orthotope facets, orthoplex vertices) and A337890 (orthotope faces, orthoplex peaks).

Table[n(1+n)(2+n)(4-3n+3n^2)/24, {n, 35}]
LinearRecurrence[{6,-15,20,-15,6,-1},{1,10,55,200,560,1316},40] (* Harvey P. Dale, Feb 15 2022 *)

a(n)=n*(n+1)*(n+2)*(3*n^2-3*n+4)/24 \\ Charles R Greathouse IV, Oct 21 2022

a(n) = n * (n+1) * (n+2) * (3*n^2 - 3*n + 4) / 24.
a(n) = 1*C(n,1) + 8*C(n,2) + 28*C(n,3) + 36*C(n,4) + 15*C(n,5), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n) = 2*A198833(n) - A047780(n) = A047780(n) - 2*A093566(n+1) = A198833(n) - A093566(n+1).
G.f.: x * (x + 4*x^2 + 10*x^3) / (1-x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Wesley Ivan Hurt, Sep 30 2020

A337953 Number of achiral colorings of the 30 edges of a regular dodecahedron or icosahedron using n or fewer colors.

Original entry on oeis.org

1, 33328, 32524281, 4312863360, 191243490675, 4239501280272, 58236754527707, 563536359633920, 4172726943804861, 25016666666700400, 126431377927701253, 554909560378102656, 2163457078062360639, 7625429483925609552, 24638829565429941975

Offset: 1

Robert A. Russell, Oct 03 2020

An achiral coloring is identical to its reflection. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual.
There are 60 elements in the automorphism group of the regular dodecahedron/icosahedron that are not in the rotation group. They divide into five conjugacy classes. The first formula is obtained by averaging the edge cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Conjugacy Class     Count    Odd Cycle Indices
  Inversion              1     x_2^15
  Edge rotation*        15     x_1^4x_2^13     Asterisk indicates that the
  Vertex rotation*      20     x_6^5           operation is followed by an
  Small face rotation*  12     x_10^3          inversion.
  Large face rotation*  12     x_10^3

Index entries for linear recurrences with constant coefficients, signature (18, -153, 816, -3060, 8568, -18564, 31824, -43758, 48620, -43758, 31824, -18564, 8568, -3060, 816, -153, 18, -1).

Cf. A282670 (oriented), A337963 (unoriented), A337964 (chiral).
Other elements: A337960 (dodecahedron vertices, icosahedron faces), A337962 (dodecahedron faces, icosahedron vertices).
Cf. A037270 (tetrahedron), A331351 (cube/octahedron).

Table[(15n^17+n^15+20n^5+24n^3)/60,{n,30}]

a(n) = n^3 * (15*n^14 + n^12 + 20*n^2 + 24) / 60.
a(n) = 1*C(n,1) + 33326*C(n,2) + 32424300*C(n,3) + 4182966200*C(n,4) + 170004083410*C(n,5) + 3156083300916*C(n,6) + 32426546302332*C(n,7) + 205938803790720*C(n,8) + 864860752435680*C(n,9) + 2503126577952000*C(n,10) + 5110943178781440*C(n,11) + 7428048096268800*C(n,12) + 7644417350169600*C(n,13) + 5446616304729600*C(n,14) + 2556525184012800*C(n,15) + 711374856192000*C(n,16) + 88921857024000*C(n,17), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n) = 2*A337963(n) - A282670(n) = A282670(n) - 2*A337964(n) = A337963(n) - A337964(n).

cp's OEIS Frontend

A060530 Number of inequivalent ways to color edges of a cube using at most n colors.

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A199406 The number of inequivalent ways to color the edges of a cube using at most n colors.

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A337414 Array read by descending antidiagonals: T(n,k) is the number of achiral colorings of the edges of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.

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A337897 Number of achiral colorings of the 8 triangular faces of a regular octahedron or the 8 vertices of a cube using n or fewer colors.

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A337406 Number of chiral pairs of colorings of the edges of a cube (or regular octahedron) using n or fewer colors.

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A337410 Array read by descending antidiagonals: T(n,k) is the number of achiral colorings of the edges of a regular n-dimensional orthotope (hypercube) using k or fewer colors.

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A337898 Number of achiral colorings of the 6 square faces of a cube or the 6 vertices of a regular octahedron using n or fewer colors.

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A337953 Number of achiral colorings of the 30 edges of a regular dodecahedron or icosahedron using n or fewer colors.

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