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

1, 8972888, 1715781087090, 9607681898535232, 7761021569825850025, 1842282666811844114760, 187827835789041358086652, 10316166994361788355074560, 353259652295786354195866209, 8333333333347091668333550200, 145411685574122353730843420174, 1978135948331715618064123517376

Offset

1,2

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.

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

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

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).

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).

Mathematica

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

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).

Sequence in context: A337964 A233492 A251157 * A234192 A162021 A157815

Adjacent sequences: A337960 A337961 A337962 * A337964 A337965 A337966

Keyword

nonn,easy

Author

Robert A. Russell, Oct 03 2020

Extensions

More terms from Andrew Howroyd, Oct 22 2025

Status

approved