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

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

Offset

1,2

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

Table of n, a(n) for n=1..9.

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) = 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).

Mathematica

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

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).

Sequence in context: A141645 A217655 A151933 * A233492 A251157 A337963

Adjacent sequences: A337961 A337962 A337963 * A337965 A337966 A337967

Keyword

nonn,easy

Author

Robert A. Russell, Oct 03 2020

Status

approved