cp's OEIS Frontend

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 1-6 of 6 results.

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

Views

Author

N. J. A. Sloane, Apr 11 2001

Keywords

Comments

Here inequivalent means under the action of the rotation group of the cube, of order 24, which in its action on the edges has cycle index (x1^12 + 3*x2^6 + 6*x4^3 + 6*x1^2*x2^5 + 8*x3^4)/24.
Also, number of inequivalent colorings of the edges of a regular octahedron using at most n colors. - José H. Nieto S., Jan 19 2012
From Robert A. Russell, Oct 08 2020: (Start)
Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbols for the regular octahedron and cube are {3,4} and {4,3} respectively. They are mutually dual.
There are 24 elements in the rotation group of the regular octahedron/cube. 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_1^12
Vertex rotation 8 x_3^4
Edge rotation 6 x_1^2x_2^5
Small face rotation 6 x_4^3
Large face rotation 3 x_2^6 (End)
Also, number of ways of coloring the vertices of the truncated tetrahedron or faces of the triakis tetrahedron up to rotation and reflection. - Peter Kagey, Nov 27 2024

References

  • 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).

Links

Crossrefs

Cf. A199406 (unoriented), A337406 (chiral), A331351 (achiral).
Other elements: A000543 (cube vertices, octahedron faces), A047780 (cube faces, octahedron vertices).
Cf. A046023 (tetrahedron), A282670 (dodecahedron/icosahedron).
Row 3 of A337407 (orthotope edges, orthoplex ridges) and A337411 (orthoplex edges, orthotope ridges).

Programs

  • Mathematica
    Table[(n^12+6n^7+3n^6+8n^4+6n^3)/24,{n,0,20}] (* Harvey P. Dale, Feb 13 2013 *)
  • PARI
    { 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

Formula

a(n) = (n^12 + 6*n^7 + 3*n^6 + 8*n^4 + 6*n^3)/24. (Replace all x_i's in the cycle index by n.)
G.f.: -x*(150*x^10 +19758*x^9 +425032*x^8 +2763481*x^7 +6769435*x^6 +6773089*x^5 +2763307*x^4 +423883*x^3 +20059*x^2 +205*x +1)/(x -1)^13. - Colin Barker, Aug 13 2012
From Robert A. Russell, Oct 08 2020: (Start)
a(n) = 1*C(n,1) + 216*C(n,2) + 22164*C(n,3) + 613804*C(n,4) + 6901425*C(n,5) + 39713430*C(n,6) + 131754420*C(n,7) + 267165360*C(n,8) + 336798000*C(n,9) + 257796000*C(n,10) + 109771200*C(n,11) + 19958400*C(n,12), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
a(n) = A199406(n) + A337406(n) = 2*A199406(n) - A331351(n) = 2*A337406(n) + A331351(n). (End)

Extensions

Entry revised by N. J. A. Sloane, Jan 03 2005

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).

A000545 Number of ways of n-coloring a dodecahedron.

Original entry on oeis.org

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

Views

Author

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

Keywords

Comments

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)

Links

Crossrefs

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).

Programs

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

Formula

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)

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

Views

Author

Vladeta Jovovic, May 20 2000

Keywords

Comments

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)

Links

Crossrefs

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).

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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

Extensions

More terms from James Sellers, May 23 2000
More terms from Colin Barker, Jul 12 2013

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).

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

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

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

Programs

  • Mathematica
    Table[(n^30-15n^17+15n^16-n^15+20n^10+24n^6-20n^5-24n^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) = 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).
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