A252704 -id:A252704 - cp's OEIS frontend

A252705 The number of ways to color the faces of a regular dodecahedron with n colors, counting mirror images as one.

1, 82, 5379, 148648, 2085655, 18356514, 116081245, 574795936, 2359033605, 8345970370, 26180606287, 74354990568, 194253329803, 472634761522, 1081541381145, 2346163937920, 4856060529001, 9641643580530, 18446420258299, 34136541925480, 61303301959263

Offset: 1

Robert A. Russell, Dec 20 2014

The cycle index using the full automorphism group for faces of a dodecahedron is (x1^12+15*x2^6+20*x3^4+24*x1^2*x5^2+15*x1^4*x2^4+x2^6+20*x6^2+24*x2*x10)/120.

Also the number of ways to color the vertices of a regular icosahedron with n colors, counting mirror images as one.

			For n=2, a(2)=82, the number of ways to color the faces of a regular dodecahedron with two colors, counting mirror images as the same. Of these, two use the same color for all faces, and 80 use both colors.

F. S. Roberts and B. Tesman, Applied Combinatorics, 2d Ed., Pearson Prentice Hall, 2005, pages 439-488.
J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge University Press, 1992, pages 461-474.

Colin Barker, 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. A000545 (number when mirror images are counted separately).

Cf. A000332 (tetrahedron), A198833 (cube), A128766 (octahedron), A252704 (icosahedron).

Table[n^2(n^2+1)(n^8-n^6+16n^4+44)/120,{n,1,30}]

vector(60, n, n^2*(n^2+1)*(n^8-n^6+16*n^4+44)/120) \\ Michel Marcus, Dec 21 2014

a(n) = n^2*(n^2+1)*(n^8-n^6+16*n^4+44)/120.
G.f.: x*(x+1)*(x^10+68*x^9+4323*x^8+80508*x^7+469548*x^6+886944*x^5+469548*x^4 +80508*x^3+4323*x^2+68*x+1)/(1-x)^13.
a(n) = C(n,1)+80*C(n,2)+5136*C(n,3)+127620*C(n,4)+1395390*C(n,5)+7965948*C(n,6) +26368272*C(n,7)+53438112*C(n,8)+67359600*C(n,9)+51559200*C(n,10)+21954240*C(n,11)+3991680*C(n,12). Each term indicates the number of ways to use n colors to color the dodecahedron with exactly 1, 2, 3, ..., 10, 11, or 12 colors.

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

Robert A. Russell, Oct 03 2020

Each chiral pair is counted as one when enumerating unoriented arrangements. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual.

Also the number of n-colorings of the vertices of the icosidodecahedron up to the 120 symmetries of the full icosahedral group. Also the number of n-colorings of faces of the rhombic triacontahedron up to the 120 symmetries of the full icosahedral group. - Peter Kagey, Sep 05 2025

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).

Cf. A282670 (oriented), A337964 (chiral), A337953 (achiral).
Other elements: A252704 (dodecahedron vertices, icosahedron faces), A252705 (dodecahedron faces, icosahedron vertices).
Other polyhedra: A063842(n-1) (tetrahedron), A199406 (cube/octahedron).

Table[(n^30+15n^17+15n^16+n^15+20n^10+24n^6+20n^5+24 n^3)/120,{n,30}]

a(n) = (n^30 + 15*n^17 + 15*n^16 + n^15 + 20*n^10 + 24*n^6 + 20*n^5 + 24*n^3) / 120.
a(n) = 1*C(n,1) + 8972886*C(n,2) + 1715754168429*C(n,3) + 9600818828024196*C(n,4) + 7713000318054315890*C(n,5) + 1795860618305879894604*C(n,6) + 175094502365510493018246*C(n,7) + 8864694277953928285823280*C(n,8) + 267022176369217557115630320*C(n,9) + 5242809910440825835898466240*C(n,10) + 71533267863142929693959229120*C(n,11) + 710438037081557065871500310400*C(n,12) + 5315930749209812373842350550400*C(n,13) + 30757743469720892095213642099200*C(n,14) + 140355611183197554763055563526400*C(n,15) + 512749946932635114150296808960000*C(n,16) + 1516429386147442831807688225280000*C(n,17) + 3659586727743885232600161343488000*C(n,18) + 7243809192262705479647976345600000*C(n,19) + 11790166608014659213935198412800000*C(n,20) + 15777861864770715186138442260480000*C(n,21) + 17309780658863308912305163714560000*C(n,22) + 15473267984805657314364466790400000*C(n,23) + 11155559298200256484274739609600000*C(n,24) + 6385716995478673633837056000000000*C(n,25) + 2834140845518322325537731379200000*C(n,26) + 939989821959452064042418176000000*C(n,27) + 219202016094796777623060480000000*C(n,28) + 32051387227306419585220608000000*C(n,29) + 2210440498434925488635904000000*C(n,30), where the coefficient of C(n,k) is the number of unoriented colorings using exactly k colors.
a(n) = A282670(n) - A337964(n) = (A282670(n) + A337953(n)) / 2 = A337964(n) + A337953(n).

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

Geoffrey Critzer, Oct 30 2011

The cycle index: 1/48 (s_1^6 + 3 s_1^4 s_2 + 9 s_1^2 s_2^2 +7 s_2^3 + 8 s_3^2 + 6 s_1^2 s_4 + 6 s_2 s_4 + 8 s_6) is returned in Mathematica by CycleIndex[ Automorphisms[ OctahedralGraph ], s].
One-sixth the area of the right triangles with sides 2b+2, b^2+2b, and b^2+2b+2 with b = A000217(n), the n-th triangular number. - J. M. Bergot, Aug 02 2013
Also the number of ways to color the faces of a cube with n colors, counting each pair of mirror images as one.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1)

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

a(n) = n*(n+1)*(n^2+n+2)*(n^2+n+4)/48.
G.f.: x*(1+3*x+7*x^2+3*x^3+x^4) / (1-x)^7. - R. J. Mathar, Oct 30 2011
a(n) = Sum_{i=1..A000217(n)} A000217(i). [Bruno Berselli, Sep 06 2013]
a(n) = 1*C(n,1) + 8*C(n,2) + 29*C(n,3) + 52*C(n,4) + 45*C(n,5) + 15*C(n,6), where the coefficient of C(n,k) is the number of unoriented colorings using exactly k colors.
a(n) = A047780(n) - A093566(n+1) = (A047780(n) + A337898(n)) / 2 = A093566(n+1) + A337898(n). - Robert A. Russell, Oct 19 2020

A054472 Number of ways to color faces of an icosahedron using at most n colors.

Original entry on oeis.org

0, 1, 17824, 58130055, 18325477888, 1589459765875, 60935989677984, 1329871177501573, 19215358684143616, 202627758536996445, 1666666669200004000, 11212499922098481787, 63895999889747261952, 316749396282749868607, 1394470923827552301472, 5542094550277768379625

Offset: 0

Vladeta Jovovic, May 20 2000

More explicitly, a(n) is the number of colorings with at most n colors of the faces of a regular icosahedron, inequivalent under the action of the rotation group of the icosahedron. It is also the number of inequivalent colorings of the vertices of a regular dodecahedron using at most n colors. - José H. Nieto S., Jan 19 2012
From Robert A. Russell, Oct 19 2020: (Start)
Each chiral pair is counted as two when enumerating oriented arrangements. 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 rotation group of the regular dodecahedron/icosahedron. They divide into five conjugacy classes. The first formula is obtained by averaging the icosahedron face (dodecahedron vertex) 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_1^20
  Vertex rotation      20     x_1^2x_3^6
  Edge rotation        15     x_2^10
  Small face rotation  12     x_5^4
  Large face rotation  12     x_5^4 (End)

Eric Weisstein's World of Mathematics, Polyhedron Coloring
Index entries for linear recurrences with constant coefficients, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).

Cf. A252704 (unoriented), A337959 (chiral), A337960 (achiral), A282670 (edges), A000545 (dodecahedron faces, icosahedron vertices), A006008 (tetrahedron), A047780 (cube faces, octahedron vertices), A000543 (octahedron faces, cube vertices).

A054472:=n->(n^20 + 15*n^10 + 20*n^8 + 24*n^4)/60; seq(A054472(n), n=0..15); # Wesley Ivan Hurt, Jan 28 2014

Table[(n^20+15n^10+20n^8+24n^4)/60,{n,0,15}] (* Harvey P. Dale, Nov 04 2011 *)
LinearRecurrence[{21,-210,1330,-5985,20349,-54264,116280,-203490,293930,-352716,352716,-293930,203490,-116280,54264,-20349,5985,-1330,210,-21,1},{0,1,17824,58130055,18325477888,1589459765875,60935989677984,1329871177501573,19215358684143616,202627758536996445,1666666669200004000,11212499922098481787,63895999889747261952,316749396282749868607,1394470923827552301472,5542094550277768379625,20148763660520129167360,67737190111299199134361,212470603607497593076128,626499557627304397693519,1747626666669235200064000},20] (* Harvey P. Dale, Aug 11 2021 *)

a(n) = (1/60)*(n^20+15*n^10+20*n^8+24*n^4).
G.f.: -x*(x +1)*(x^18 +17802*x^17 +57738159*x^16 +17050750284*x^15 +1199757591558*x^14 +30128721042672*x^13 +329847884196810*x^12 +1749288479932404*x^11 +4727182539811968*x^10 +6598854419308684*x^9 +4727182539811968*x^8 +1749288479932404*x^7 +329847884196810*x^6 +30128721042672*x^5 +1199757591558*x^4 +17050750284*x^3 +57738159*x^2 +17802*x +1) / (x -1)^21. - Colin Barker, Jul 13 2013
a(n) = 1*C(n,1) + 17822*C(n,2) + 58076586*C(n,3) + 18093064608*C(n,4) + 1498413498750*C(n,5) + 51672950917308*C(n,6) + 936058547290608*C(n,7) + 10194866756893728*C(n,8) + 72644237439379200*C(n,9) + 357895538663241600*C(n,10) + 1264592451488446080*C(n,11) + 3281293750348373760*C(n,12) + 6337930306906598400*C(n,13) + 9157388718839961600*C(n,14) + 9858321678965760000*C(n,15) + 7794071905639219200*C(n,16) + 4394429252269056000*C(n,17) + 1672620130621440000*C(n,18) + 385209484627968000*C(n,19) + 40548366802944000*C(n,20), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors. - Robert A. Russell, Dec 03 2014
a(n) = A252704(n) + A337959(n) = 2*A252704(n) - A337960(n) = 2*A337959(n) + A337960(n). - Robert A. Russell, Oct 19 2020

More terms from James Sellers, May 23 2000

More terms from Colin Barker, Jul 12 2013

A337960 Number of achiral colorings of the 30 triangular faces of a regular icosahedron or the 30 vertices of a regular dodecahedron using n or fewer colors.

Original entry on oeis.org

1, 1048, 133875, 4211872, 61198135, 545203800, 3465030541, 17197766272, 70665499413, 250166670040, 785039389519, 2230057075104, 5826818931739, 14178299017624, 32446195329465, 70387069393408, 145689159233737

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 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_2^10
  Edge rotation*        15     x_1^4x_2^8      Asterisk indicates that the
  Vertex rotation*      20     x_2^1x_6^3      operation is followed by an
  Small face rotation*  12     x_10^2          inversion.
  Large face rotation*  12     x_10^2

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

Cf. A054472 (oriented), A252704 (unoriented), A337959 (chiral).
Other elements: A337953 (edges), A337962 (dodecahedron faces, icosahedron vertices).
Other polyhedra: A006003 (tetrahedron), A337898 (cube faces, octahedron vertices), A337897 (octahedron faces, cube vertices).

Table[(15n^12+n^10+20n^4+24n^2)/60,{n,30}]

a(n) = n^2 * (15*n^10 + n^8 + 20*n^2 + 24) / 60.
a(n) = 1*C(n,1) + 1046*C(n,2) + 130734*C(n,3) + 3682656*C(n,4) + 41467050*C(n,5) + 238531284*C(n,6) + 791012880*C(n,7) + 1603496160*C(n,8) + 2021060160*C(n,9) + 1546836480*C(n,10) + 658627200*C(n,11) + 119750400*C(n,12), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n) = 2*A252704(n) - A054472(n) = A054472(n) - 2*A337959(n) = A252704(n) - A337959(n).

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

Peter Kagey, Nov 27 2024

nonn

Equivalently, the number of n-colorings of the faces of the tetrakis hexahedron, which is the polyhedral dual of the truncated octahedron.

Colorings are counted up to the full octahedral group of order 48.

Wikipedia, Tetrakis hexahedron
Wikipedia, Truncated octahedron

Cf. A000332, A060530, A128766, A199406, A252704, A252705, A274900, A337963, A378474, A378475, A378476, A378477, A378478.

a(n) = (1/48)*(n^24 + 3*n^16 + 16*n^12 + 8*n^8 + 12*n^6 + 8*n^4).

Asymptotically, a(n) ~ n^24/48.

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

Peter Kagey, Nov 27 2024

Equivalently, the number of n-colorings of the faces of the disdyakis dodecahedron, which is the polyhedral dual of the truncated cuboctahedron.

Colorings are counted up to the full octahedral group of order 48.

Wikipedia, Disdyakis dodecahedron
Wikipedia, Truncated cuboctahedron

Cf. A000332, A060530, A128766, A199406, A252704, A252705, A274900, A337963, A378473, A378475, A378476, A378477, A378478.

a(n) = (1/48)*(n^48 + 19*n^24 + 8*n^16 + 12*n^12 + 8*n^8).

Asymptotically, a(n) ~ n^48/48.

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

Peter Kagey, Nov 27 2024

nonn

Equivalently, the number of n-colorings of the faces of the pentagonal icositetrahedron, which is the polyhedral dual of the snub cube.
Colorings are counted up to the rotational octahedral symmetry group of order 24.
This is also:
1) The number of n-colorings of the vertices of the truncated octahedron (equivalently faces of the tetrakis hexahedron) up to rotational octahedral symmetry (alternatively full tetrahedral symmetry).
2) The number of n-colorings of the vertices of the truncated cube (equivalently faces of the triakis octahedron) up to rotational octahedral symmetry.

Wikipedia, Pentagonal icositetrahedron
Wikipedia, Snub cube

Cf. A000332, A060530, A128766, A199406, A252704, A252705, A274900, A337963, A378473, A378474, A378476, A378477, A378478.

a(n) = (1/24)*(n^24 + 9*n^12 + 8*n^8 + 6*n^6).

Asymptotically, a(n) ~ n^24/24.

A378476 The number of n-colorings of the vertices of the truncated dodecahedron up to rotation and reflection.

Original entry on oeis.org

0, 1, 9607679885269312, 353259652293727442874919719, 11076899964874301400431118585745408, 7228014483236696229750911410649667971875, 407280649839077145745380578110103790290896704, 4233515506163528044351709372473136729199352546645

Offset: 0

Peter Kagey, Nov 27 2024

Equivalently,
1) the number of n-colorings of the faces of the triakis icosahedron, which is the polyhedral dual of the truncated dodecahedron.
2) the number of n-colorings of the faces of the pentakis dodecahedron, or n-colorings of the vertices of the truncated icosahedron, its polyhedral dual.
3) the number of n-colorings of the faces of the deltoidal hexecontahedron, or n-colorings of the vertices of the rhombicosidodecahedron, its polyhedral dual.
Colorings are counted up to the full icosahedral symmetry group of order 120.

Cf. A000332, A060530, A128766, A199406, A252704, A252705, A274900, A337963, A378473, A378474, A378475, A378477, A378478.

a(n) = (1/120)*(n^60 + 15*n^32 + 16*n^30 + 20*n^20 + 24*n^12 + 20*n^10 + 24*n^6).

Asymptotically, a(n) ~ n^60/120.

a(0) = 0 prepended by Georg Fischer, Apr 16 2025

A378477 The number of n-colorings of the vertices of the truncated icosidodecahedron up to rotation and reflection.

Original entry on oeis.org

0, 1, 11076899964874299238703297447907328, 14975085832620260086776498590197757887552760437584786915, 14723725539819869413194145839524321308612931385268246121155792029614080, 6269303204385533375833261531851976948366440371233447120478861810030555725146484375

Offset: 0

Peter Kagey, Nov 27 2024

nonn

Equivalently, the number of n-colorings of the faces of the disdyakis triacontahedron, which is the polyhedral dual of the truncated octahedron.

Colorings are counted up to the full icosahedral symmetry group of order 120.

Wikipedia, Disdyakis triacontahedron
Wikipedia, Truncated icosidodecahedron

Cf. A000332, A060530, A128766, A199406, A252704, A252705, A274900, A337963, A378473, A378474, A378475, A378476, A378478.

a(n) = 1/120*(n^120 + 31*n^60 + 20*n^40 + 24*n^24 + 20*n^20 + 24*n^12).

Asymptotically, a(n) ~ n^120/120

cp's OEIS Frontend

A252705 The number of ways to color the faces of a regular dodecahedron with n colors, counting mirror images as one.

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A337963 Number of unoriented colorings of the 30 edges of a regular dodecahedron or icosahedron using n or fewer colors.

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A198833 The number of inequivalent ways to color the vertices of a regular octahedron using at most n colors.

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Magma

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A054472 Number of ways to color faces of an icosahedron using at most n colors.

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A337960 Number of achiral colorings of the 30 triangular faces of a regular icosahedron or the 30 vertices of a regular dodecahedron using n or fewer colors.

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A378473 The number of n-colorings of the vertices of the truncated octahedron up to rotation and reflection.

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A378474 The number of n-colorings of the vertices of the truncated cuboctahedron up to rotation and reflection.

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A378475 The number of n-colorings of the vertices of the snub cube up to rotation.

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