|
|
A337954
|
|
Number of chiral pairs of colorings of the 16 tetrahedral facets of a hyperoctahedron or of the 16 vertices of a tesseract.
|
|
8
|
|
|
0, 94, 97974, 10700090, 390081800, 7280687610, 86121007714, 730895668104, 4816861200630, 26010740238450, 119563513291420, 481192778757834, 1732132086737234, 5669991002636870, 17101193825828700, 48029634770843680
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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 tesseract and the hyperoctahedron are {4,3,3} and {3,3,4} respectively. Both figures are regular 4-D polyhedra and they are mutually dual.
|
|
LINKS
|
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).
|
|
FORMULA
|
a(n) = (n-1) * n^2 * (n+1) * (n^12 + n^10 - 11*n^8 + n^6 + 44 n^4 - 4 n^2 - 48) / 384.
a(n) = 94*C(n,2) + 97692*C(n,3) + 10308758*C(n,4) + 337560150*C(n,5) + 5098740090*C(n,6) + 42976836210*C(n,7) + 224685801060*C(n,8) + 775389028050*C(n,9) + 1830791421900*C(n,10) + 3007909258200*C(n,11) + 3439214024400*C(n,12) + 2685727044000*C(n,13) + 1366701336000*C(n,14) + 408648240000*C(n,15) + 54486432000*C(n,16), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
|
|
MATHEMATICA
|
Table[(n^16-12n^12+12n^10+43n^8-48n^6-44n^4+48n^2)/384, {n, 30}]
|
|
CROSSREFS
|
Other elements: A331360 (tesseract edges, hyperoctahedron faces), A331356 (tesseract faces, hyperoctahedron edges), A234249(n+1) (tesseract facets, hyperoctahedron vertices).
Row 4 of A325014 (orthoplex facets, orthotope vertices).
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|