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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Robert A. Russell, Oct 03 2020

Keywords

Comments

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.

Links

  • 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).

Crossrefs

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).

Programs

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

Formula

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).

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

Views

Author

David Nacin, Feb 20 2017

Keywords

Comments

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)

Examples

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

Links

  • 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).

Crossrefs

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).

Programs

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

Formula

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

Views

Author

Robert A. Russell, Oct 03 2020

Keywords

Comments

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

Links

Crossrefs

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).

Programs

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

Formula

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).

A337961 Number of chiral pairs of colorings of the 12 pentagonal faces of a regular dodecahedron or the 12 vertices of a regular icosahedron using n or fewer colors.

Original entry on oeis.org

0, 14, 3720, 132184, 1987720, 17935806, 114638048, 570597216, 2348263008, 8320953630, 26126986952, 74247445272, 194049316552, 472265688622, 1080900468480, 2345089916288, 4854316187136, 9638888023278, 18442173583176
Offset: 1

Views

Author

Robert A. Russell, Oct 03 2020

Keywords

Comments

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.

Links

Crossrefs

Cf. A000545 (oriented), A252705 (unoriented), A337962 (achiral).
Other elements: A337959 (dodecahedron vertices, icosahedron faces), A337964 (edges).
Other polyhedra: A000332 (tetrahedron), A093566(n+1) (cube faces, octahedron vertices), A337896 (octahedron faces, cube vertices).

Programs

  • Mathematica
    Table[(n^12-15n^8+14n^6+44n^4-44n^2)/120,{n,30}]
LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{0,14,3720,132184,1987720,17935806,114638048,570597216,2348263008,8320953630,26126986952,74247445272,194049316552},20] (* Harvey P. Dale, Nov 17 2024 *)

Formula

a(n) = (n-1) * n^2 * (n+1) * (n^8 + n^6 - 14*n^4 + 44) / 120.
a(n) = 14*C(n,2) + 3678*C(n,3) + 117388*C(n,4) + 1363860*C(n,5) + 7918056*C(n,6) + 26332992*C(n,7) + 53428032*C(n,8) + 67359600*C(n,9) + 51559200*C(n,10) + 21954240*C(n,11) + 3991680*C(n,12), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
a(n) = A000545(n) - A252705(n) = (A000545(n) - A337962(n)) / 2 = A252705(n) - A337962(n).

A337959 Number of chiral pairs of 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

0, 8388, 28998090, 9160633008, 794699283870, 30467722237092, 664933856235516, 9607670743188672, 101313843935748516, 833333209516666980, 5606249568529546134, 31947998829845093424, 158374695227965468434
Offset: 1

Views

Author

Robert A. Russell, Oct 03 2020

Keywords

Comments

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.

Links

Crossrefs

Cf. A054472 (oriented), A252704 (unoriented), A337960 (achiral).
Other elements: A337964 (edges), A337961 (dodecahedron faces, icosahedron vertices).
Other polyhedra: A000332 (tetrahedron), A093566(n+1) (cube faces, octahedron vertices), A337896 (octahedron faces, cube vertices).

Programs

  • Mathematica
    Table[(n^20-15n^12+14n^10+20n^8+4n^4-24n^2)/120,{n,30}]

Formula

a(n) = (n-1) * n^2 * (n+1) * (n^2+2) * (n^14 - n^12 + 3*n^10 - 5*n^8 - 4*n^6 + 8*n^4 + 4*n^2 + 12) /120.
a(n) = 8388*C(n,2) + 28972926*C(n,3) + 9044690976*C(n,4) + 749186015850*C(n,5) + 25836356193012*C(n,6) + 468028878138864*C(n,7) + 5097432576698784*C(n,8) + 36322117709159520*C(n,9) + 178947768558202560*C(n,10) + 632296225414909440*C(n,11) + 1640646875114311680*C(n,12) + 3168965153453299200*C(n,13) + 4578694359419980800*C(n,14) + 4929160839482880000*C(n,15) + 3897035952819609600*C(n,16) + 2197214626134528000*C(n,17) + 836310065310720000*C(n,18) + 192604742313984000*C(n,19) + 20274183401472000*C(n,20), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
a(n) = A054472(n) - A252704(n) = (A054472(n) - A337960(n)) / 2 = A252704(n) - A337960(n).
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