A198833 The number of inequivalent ways to color the vertices of a regular octahedron using at most n colors.

1, 10, 56, 220, 680, 1771, 4060, 8436, 16215, 29260, 50116, 82160, 129766, 198485, 295240, 428536, 608685, 848046, 1161280, 1565620, 2081156, 2731135, 3542276, 4545100, 5774275, 7268976, 9073260, 11236456, 13813570, 16865705, 20460496, 24672560, 29583961

Offset: 1

Author

Geoffrey Critzer, Oct 30 2011

Keywords

• nonn
• easy

Comments

The cycle index: 1/48 (s_1^6 + 3 s_1^4 s_2 + 9 s_1^2 s_2^2 +7 s_2^3 + 8 s_3^2 + 6 s_1^2 s_4 + 6 s_2 s_4 + 8 s_6) is returned in Mathematica by CycleIndex[ Automorphisms[ OctahedralGraph ], s].

One-sixth the area of the right triangles with sides 2b+2, b^2+2b, and b^2+2b+2 with b = A000217(n), the n-th triangular number. - J. M. Bergot, Aug 02 2013

Also the number of ways to color the faces of a cube with n colors, counting each pair of mirror images as one.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1)

Crossrefs

Cf. A047780 (oriented), A093566(n+1) (chiral), A337898 (achiral), A199406 (edges), A128766 (octahedron faces, cube vertices), A000332(n+3) (tetrahedron), A128766 (octahedron faces, cube vertices), A252705 (dodecahedron faces, icosahedron vertices), A252704 (icosahedron faces, dodecahedron vertices), A000217 (triangular numbers).

Row 3 of A325005 (orthotope facets, orthoplex vertices) and A337888 (orthotope faces, orthoplex peaks).

Programs

Magma

[n*(n+1)*(n^2+n+2)*(n^2+n+4)/48: n in [1..35]]; // Vincenzo Librandi, Aug 04 2013

Mathematica

Table[(n^6 + 3 n^5 + 9 n^4 + 13 n^3 + 14 n^2 + 8 n)/48, {n, 25}]
CoefficientList[Series[-(1 + 3 x + 7 x^2 + 3 x^3 + x^4) / (x - 1)^7, {x, 0, 35}], x] (* Vincenzo Librandi, Aug 04 2013 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,10,56,220,680,1771,4060},40] (* Harvey P. Dale, Nov 06 2024 *)

PARI

a(n)=n*(n+1)*(n^2+n+2)*(n^2+n+4)/48 \\ Charles R Greathouse IV, Aug 02 2013

Formula

a(n) = n*(n+1)*(n^2+n+2)*(n^2+n+4)/48.

G.f.: x*(1+3*x+7*x^2+3*x^3+x^4) / (1-x)^7. - R. J. Mathar, Oct 30 2011

a(n) = Sum_{i=1..A000217(n)} A000217(i). [Bruno Berselli, Sep 06 2013]

a(n) = 1*C(n,1) + 8*C(n,2) + 29*C(n,3) + 52*C(n,4) + 45*C(n,5) + 15*C(n,6), where the coefficient of C(n,k) is the number of unoriented colorings using exactly k colors.

a(n) = A047780(n) - A093566(n+1) = (A047780(n) + A337898(n)) / 2 = A093566(n+1) + A337898(n). - Robert A. Russell, Oct 19 2020