A337961 -id:A337961 - cp's OEIS frontend

A000545 Number of ways of n-coloring a dodecahedron.

1, 96, 9099, 280832, 4073375, 36292320, 230719293, 1145393152, 4707296613, 16666924000, 52307593239, 148602435840, 388302646355, 944900450144, 2162441849625, 4691253854208, 9710376716137, 19280531603808, 36888593841475, 68266682784000, 122597146773927

Offset: 1

Clint. C. Williams (Clintwill(AT)aol.com)

More explicitly, a(n) is the number of colorings with at most n colors of the faces of a regular dodecahedron, inequivalent under the action of the rotation group of the dodecahedron. It is also the number of inequivalent colorings of the vertices of a regular icosahedron using at most n colors. - José H. Nieto S., Jan 19 2012
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 dodecahedron face (icosahedron 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_1^12
  Edge rotation        15     x_2^6
  Vertex rotation      20     x_3^4
  Small face rotation  12     x_1^2x_5^2
  Large face rotation  12     x_1^2x_5^2  (End)

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Polyhedron Coloring
Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Cf. A252705 (unoriented), A337961 (chiral), A337962 (achiral).
Other elements: A054472 (dodecahedron vertices, icosahedron faces), A282670 (edges).
Other polyhedra: A006008 (tetrahedron), A047780 (cube faces, octahedron vertices), A000543 (octahedron faces, cube vertices).

Maple
```
(1/60)*n^12+(1/4)*n^6+(11/15)*n^4;
```

Table[n^12/60+n^6/4+11 n^4/15,{n,20}] (* or *) CoefficientList[Series[ -(((1+x) (1+x (82+x (7847+x (161900+x (943640+x (1764740+x (943640+x (161900+x (7847+x (82+x)))))))))))/(x-1)^13),{x,0,20}],x] (* Harvey P. Dale, Apr 25 2011 *)

G.f.: x*((1+x)*(1+x*(82+x*(7847+x*(161900+x*(943640+x*(1764740+x*(943640+x*(161900+x*(7847+x*(82+x)))))))))))/(1-x)^13. - Harvey P. Dale, Apr 25 2011
From Robert A. Russell, Oct 03 2020: (Start)
a(n) = (n^12 + 15*n^6 + 44*n^4) / 60.
a(n) = 1*C(n,1) + 94*C(n,2) + 8814*C(n,3) + 245008*C(n,4) + 2759250*C(n,5) + 15884004*C(n,6) + 52701264*C(n,7) + 106866144*C(n,8) + 134719200*C(n,9) + 103118400*C(n,10) + 43908480*C(n,11) + 7983360*C(n,12), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
a(n) = A252705(n) + A337961(n) = 2*A252705(n) - A337962(n) = 2*A337961(n) + A337962(n). (End)

A337962 Number of achiral 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

1, 68, 1659, 16464, 97935, 420708, 1443197, 4198720, 10770597, 25016740, 53619335, 107545296, 204013251, 369072900, 640912665, 1074021632, 1744341865, 2755557252, 4246675123, 6401066960, 9457144599, 13720858404

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 dodecahedron face (icosahedron 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_2^6
  Edge rotation*        15     x_1^4x_2^4      Asterisk indicates that the
  Vertex rotation*      20     x_6^2           operation is followed by an
  Small face rotation*  12     x_2^1x_10^1     inversion.
  Large face rotation*  12     x_2^1x_10^1

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000545 (oriented), A252705 (unoriented), A337961 (chiral).
Other elements: A337960 (dodecahedron vertices, icosahedron faces), A337953 (edges).
Other polyhedra: A006003 (tetrahedron), A337898 (cube faces, octahedron vertices), A337897 (octahedron faces, cube vertices).

Mathematica
```
Table[(15n^8+n^6+44n^2)/60,{n,30}]
```

a(n) = n^2 * (15*n^6 + n^4 + 44)/60.
a(n) = 1*C(n,1) + 66*C(n,2) + 1458*C(n,3) + 10232*C(n,4) + 31530*C(n,5) + 47892*C(n,6) + 35280*C(n,7) + 10080*C(n,8), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n) = 2*A252705(n) - A000545(n) = A000545(n) - 2*A337961(n) = A252705(n) - A337961(n).
From Stefano Spezia, Oct 04 2020: (Start)
G.f.: x*(1+59*x+1083*x^2+3897*x^3+3087*x^4+1083*x^5+59*x^6+x^7)/(1-x)^9.
a(n) = 9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-8) for n > 8.
(End)

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).

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

Robert A. Russell, Oct 03 2020

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. 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).

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

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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A000545 Number of ways of n-coloring a dodecahedron.

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A337962 Number of achiral colorings of the 12 pentagonal faces of a regular dodecahedron or the 12 vertices of a regular icosahedron using n or fewer colors.

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

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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.

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