A060530
Number of inequivalent ways to color edges of a cube using at most n colors.
Original entry on oeis.org
0, 1, 218, 22815, 703760, 10194250, 90775566, 576941778, 2863870080, 11769161895, 41669295250, 130772947481, 371513523888, 970769847320, 2362273657030, 5406141568500, 11728193258496, 24276032182173, 48201464902410, 92221684354915
Offset: 0
- 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).
Other elements:
A000543 (cube vertices, octahedron faces),
A047780 (cube faces, octahedron vertices).
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
A337963
Number of unoriented colorings of the 30 edges of a regular dodecahedron or icosahedron using n or fewer colors.
Original entry on oeis.org
1, 8972888, 1715781087090, 9607681898535232, 7761021569825850025, 1842282666811844114760, 187827835789041358086652, 10316166994361788355074560, 353259652295786354195866209
Offset: 1
- Index entries for linear recurrences with constant coefficients, signature (31, -465, 4495, -31465, 169911, -736281, 2629575, -7888725, 20160075, -44352165, 84672315, -141120525, 206253075, -265182525, 300540195, -300540195, 265182525, -206253075, 141120525, -84672315, 44352165, -20160075, 7888725, -2629575, 736281, -169911, 31465, -4495, 465, -31, 1).
Other elements:
A252704 (dodecahedron vertices, icosahedron faces),
A252705 (dodecahedron faces, icosahedron vertices).
Other polyhedra:
A063842(n-1) (tetrahedron),
A199406 (cube/octahedron).
A198833
The number of inequivalent ways to color the vertices of a regular octahedron using at most n colors.
Original entry on oeis.org
1, 10, 56, 220, 680, 1771, 4060, 8436, 16215, 29260, 50116, 82160, 129766, 198485, 295240, 428536, 608685, 848046, 1161280, 1565620, 2081156, 2731135, 3542276, 4545100, 5774275, 7268976, 9073260, 11236456, 13813570, 16865705, 20460496, 24672560, 29583961
Offset: 1
Cf.
A047780 (oriented),
A093566(n+1) (chiral),
A337898 (achiral),
A199406 (edges),
A128766 (octahedron faces, cube vertices),
A000332(n+3) (tetrahedron),
A128766 (octahedron faces, cube vertices),
A252705 (dodecahedron faces, icosahedron vertices),
A252704 (icosahedron faces, dodecahedron vertices),
A000217 (triangular numbers).
Row 3 of
A325005 (orthotope facets, orthoplex vertices) and
A337888 (orthotope faces, orthoplex peaks).
-
[n*(n+1)*(n^2+n+2)*(n^2+n+4)/48: n in [1..35]]; // Vincenzo Librandi, Aug 04 2013
-
Table[(n^6 + 3 n^5 + 9 n^4 + 13 n^3 + 14 n^2 + 8 n)/48, {n, 25}]
CoefficientList[Series[-(1 + 3 x + 7 x^2 + 3 x^3 + x^4) / (x - 1)^7, {x, 0, 35}], x] (* Vincenzo Librandi, Aug 04 2013 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,10,56,220,680,1771,4060},40] (* Harvey P. Dale, Nov 06 2024 *)
-
a(n)=n*(n+1)*(n^2+n+2)*(n^2+n+4)/48 \\ Charles R Greathouse IV, Aug 02 2013
A337412
Array read by descending antidiagonals: T(n,k) is the number of unoriented 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, 21, 144, 1, 5, 55, 12111, 49127, 1, 6, 120, 358120, 740360358, 293122232, 1, 7, 231, 5131650, 733776248840, 3168520600399659, 25174334733080, 1, 8, 406, 45528756, 155261523065875, 314848558732420555904, 920040738175691418086226, 30035285091978202824, 1
Offset: 1
Table begins with T(1,1):
1 2 3 4 5 6 7 8 9 ...
1 6 21 55 120 231 406 666 1035 ...
1 144 12111 358120 5131650 45528756 288936634 1433251296 5887880415 ...
For T(2,2)=6, the arrangements are AAAA, AAAB, AABB, ABAB, ABBB, and BBBB.
-
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; 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)]
array[n_, k_] := row[n] /. b -> k
Table[array[n,d+m-n], {d,8}, {n,m,d+m-1}] // Flatten
A331351
Number of achiral colorings of the edges of a cube or regular octahedron.
Original entry on oeis.org
1, 70, 1407, 12480, 69050, 281946, 931490, 2632512, 6598935, 15041950, 31740841, 62830560, 117855192, 211141490, 363551700, 604679936, 975561405, 1531968822, 2348375395, 3522668800, 5181705606, 7487800650, 10646250902
Offset: 1
- G. Royle, Partitions and Permutations
- Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).
Row 3 of
A337410 (orthotope edges, orthoplex ridges) and
A337414 (orthoplex edges, orthotope ridges).
-
Table[(8n^2 + 6n^3 + n^6 + 6n^7 + 3n^8)/24, {n, 1, 30}]
LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {1, 70, 1407, 12480, 69050, 281946, 931490, 2632512, 6598935}, 25]
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
- Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).
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 *)
A337408
Array read by descending antidiagonals: T(n,k) is the number of unoriented 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, 21, 144, 1, 5, 55, 12111, 11251322, 1, 6, 120, 358120, 4825746875682, 314824456456819827136, 1, 7, 231, 5131650, 48038446526132256, 38491882660019692002988737797054040, 136221825854745676520058554256163406987047485113810944, 1
Offset: 1
Table begins with T(1,1):
1 2 3 4 5 6 7 8 9 ...
1 6 21 55 120 231 406 666 1035 ...
1 144 12111 358120 5131650 45528756 288936634 1433251296 5887880415 ...
For T(2,2)=6, the arrangements are AAAA, AAAB, AABB, ABAB, ABBB, and BBBB.
-
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; 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)]
array[n_, k_] := row[n] /. b -> k
Table[array[n,d+m-n], {d,7}, {n,m,d+m-1}] // Flatten
A378473
The number of n-colorings of the vertices of the truncated octahedron up to rotation and reflection.
Original entry on oeis.org
0, 1, 355048, 5886817533, 5864336054656, 1241773051013125, 98716454926955496, 3991277735434713913, 98382652674879674368, 1661801013342756245961, 20833333958666683585000, 205202766952229526577141, 1656184328295547539616128, 11308349424395689922231053
Offset: 0
Cf.
A000332,
A060530,
A128766,
A199406,
A252704,
A252705,
A274900,
A337963,
A378474,
A378475,
A378476,
A378477,
A378478.
A378474
The number of n-colorings of the vertices of the truncated cuboctahedron up to rotation and reflection.
Original entry on oeis.org
0, 1, 5864068667776, 1661800897546646288751, 1650586719047285117763813376, 74014868308343792955106160546875, 467755368903219944377426648894114176, 764653504526960946768130306131125170501, 464598858302721315450530067459906444722176
Offset: 0
Cf.
A000332,
A060530,
A128766,
A199406,
A252704,
A252705,
A274900,
A337963,
A378473,
A378475,
A378476,
A378477,
A378478.
A378475
The number of n-colorings of the vertices of the snub cube up to rotation.
Original entry on oeis.org
0, 1, 700688, 11768099013, 11728130343936, 2483526957328125, 197432556580265616, 7982551312716034313, 196765270145344012288, 3323601794975613468921, 41666666667041700250000, 410405528159827444816781, 3312368633477962187301888, 22616698765607508420521013
Offset: 0
Cf.
A000332,
A060530,
A128766,
A199406,
A252704,
A252705,
A274900,
A337963,
A378473,
A378474,
A378476,
A378477,
A378478.
Showing 1-10 of 15 results.
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