A337963 -id:A337963 - cp's OEIS frontend

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

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)

A282670 Number of inequivalent ways to color the edges of a dodecahedron using at most n colors.

Original entry on oeis.org

0, 1, 17912448, 3431529649899, 19215359484207104, 15522042948408209375, 3684565329384186949248, 375655671519845961645597, 20632333988160040350515200, 706519304587399981447927557, 16666666666669166670000400000, 290823371148118276083759139095

Offset: 0

David Nacin, Feb 20 2017

Cycle index of symmetry group A5 acting on the 30 edges of the dodecahedron is (24s(5)^6 + 20s(3)^10 + 15s(2)^14*s(1)^2 + s(1)^30)/60.
Also the number of inequivalent ways to color the edges of the icosahedron using at most n colors.
From Robert A. Russell, Oct 03 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 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_1^30
  Edge rotation        15     x_1^2x_2^14
  Vertex rotation      20     x_3^10
  Small face rotation  12     x_5^6
Large face rotation  12     x_5^6  (End)

			There are a(2) = 17912448 inequivalent ways to color the edges of the dodecahedron using at most two colors.

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: A054472 (dodecahedron vertices, icosahedron faces), A000545 (dodecahedron faces, icosahedron vertices).
Other polyhedra: A046023 (tetrahedron), A060530 (cube/octahedron).
Cf. A337963 (unoriented), A337964 (chiral), A337953 (achiral).

Table[(24n^6+20n^10+15n^16+n^30)/60, {n, 0, 16}]

a(n) = n^6 (n^24 + 15 n^10 + 20 n^4 + 24)/60.
G.f.: x*(1 + x)*(1 + 17912416*x + 3430956452060*x^2 + 19105559437892000*x^3 + 14908856825730677891*x^4 + 3197392859155796794496*x^5 + 265368238349945588707496*x^6 + 10365795256050146806088576*x^7 + 215154060506484358838662001*x^8 + 2568188846096433625477331936*x^9 + 18582986600475456162494990756*x^10 + 84400699070086923625163495456*x^11 + 245956255494355672481225103371*x^12 + 465612713610802763378946154496*x^13 + 575747234318647571242943474096*x^14 + 465612713610802763378946154496*x^15 + 245956255494355672481225103371*x^16 + 84400699070086923625163495456*x^17 + 18582986600475456162494990756*x^18 + 2568188846096433625477331936*x^19 + 215154060506484358838662001*x^20 + 10365795256050146806088576*x^21 + 265368238349945588707496*x^22 + 3197392859155796794496*x^23 + 14908856825730677891*x^24 + 19105559437892000*x^25 + 3430956452060*x^26 + 17912416*x^27 + x^28) / (1 - x)^31. - Colin Barker, Mar 30 2019
From Robert A. Russell, Oct 03 2020: (Start)
a(n) = 1*C(n,1) + 17912446*C(n,2) + 3431475912558*C(n,3) + 19201633473082192*C(n,4) + 15426000466104548370*C(n,5) + 3591721233455676488292*C(n,6) + 350189004698594439734160*C(n,7) + 17729388555701917767855840*C(n,8) + 534044352737570253478824960*C(n,9) + 10485619820879148545218980480*C(n,10) + 143066535726280748444739676800*C(n,11) + 1420876074163106703694904352000*C(n,12) + 10631861498419617103267350931200*C(n,13) + 61515486939441778743810979468800*C(n,14) + 280711222366395106969585943040000*C(n,15) + 1025499893865270227589218761728000*C(n,16) + 3032858772294885663526454593536000*C(n,17) + 7319173455487770465200322686976000*C(n,18) + 14487618384525410959295952691200000*C(n,19) + 23580333216029318427870396825600000*C(n,20) + 31555723729541430372276884520960000*C(n,21) + 34619561317726617824610327429120000*C(n,22) + 30946535969611314628728933580800000*C(n,23) + 22311118596400512968549479219200000*C(n,24) + 12771433990957347267674112000000000*C(n,25) + 5668281691036644651075462758400000*C(n,26) + 1879979643918904128084836352000000*C(n,27) + 438404032189593555246120960000000*C(n,28) + 64102774454612839170441216000000*C(n,29) + 4420880996869850977271808000000*C(n,30), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
a(n) = A337963(n) + A337964(n) = 2*A337963(n) - A337953(n) = 2*A337964(n) + A337953(n). (End)

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).

A337964 Number of chiral pairs of colorings of the 30 edges of a regular dodecahedron or icosahedron using n or fewer colors.

Original entry on oeis.org

0, 8939560, 1715748562809, 9607677585671872, 7761021378582359350, 1842282662572342834488, 187827835730804603558945, 10316166993798251995440640, 353259652291613627252061348

Offset: 1

Robert A. Russell, Oct 03 2020

Each member of a chiral pair is a reflection, but not a rotation, of the other. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual.

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), A337963 (unoriented), A337953 (achiral).
Other elements: A337959 (dodecahedron vertices, icosahedron faces), A337961 (dodecahedron faces, icosahedron vertices).
Other polyhedra: A337899 (tetrahedron), A337406 (cube/octahedron).

Table[(n^30-15n^17+15n^16-n^15+20n^10+24n^6-20n^5-24n^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) = 8939560*C(n,2) + 1715721744129*C(n,3) + 9600814645057996*C(n,4) + 7713000148050232480*C(n,5) + 1795860615149796593688*C(n,6) + 175094502333083946715914*C(n,7) + 8864694277747989482032560*C(n,8) + 267022176368352696363194640*C(n,9) + 5242809910438322709320514240*C(n,10) + 71533267863137818750780447680*C(n,11) + 710438037081549637823404041600*C(n,12) + 5315930749209804729425000380800*C(n,13) + 30757743469720886648597337369600*C(n,14) + 140355611183197552206530379513600*C(n,15) + 512749946932635113438921952768000*C(n,16) + 1516429386147442831718766368256000*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 chiral pairs of colorings using exactly k colors.
a(n) = A282670(n) - A337963(n) = (A282670(n) - A337953(n)) / 2 = A337963(n) - A337953(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

A378478 The number of n-colorings of the vertices of the snub dodecahedron up to rotation.

Original entry on oeis.org

0, 1, 19215358678900736, 706519304586988199183738259, 22153799929748598169960860333637632, 14456028966473392453665534687042333984375, 814561299678154291488767806377392301451223040, 8467031012327056088703142262372040966699399765293

Offset: 0

Peter Kagey, Nov 27 2024

nonn

Equivalently, the number of n-colorings of the faces of the pentagonal hexecontahedron, which is the polyhedral dual of the snub dodecahedron.

Colorings are counted up to the rotational icosahedral symmetry group of order 60.

Wikipedia, Pentagonal hexecontahedron
Wikipedia, Snub dodecahedron

Cf. A000332, A060530, A128766, A199406, A252704, A252705, A274900, A337963, A378473, A378474, A378475, A378476, A378477.

a(n) = 1/60*(n^60 + 15*n^30 + 20*n^20 + 24*n^12).

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A282670 Number of inequivalent ways to color the edges of a dodecahedron using at most n colors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A337964 Number of chiral pairs of colorings of the 30 edges of a regular dodecahedron or icosahedron using n or fewer colors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments