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%I A198833 #55 Nov 06 2024 11:17:06
%S A198833 1,10,56,220,680,1771,4060,8436,16215,29260,50116,82160,129766,198485,
%T A198833 295240,428536,608685,848046,1161280,1565620,2081156,2731135,3542276,
%U A198833 4545100,5774275,7268976,9073260,11236456,13813570,16865705,20460496,24672560,29583961
%N A198833 The number of inequivalent ways to color the vertices of a regular octahedron using at most n colors.
%C A198833 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].
%C A198833 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
%C A198833 Also the number of ways to color the faces of a cube with n colors, counting each pair of mirror images as one.
%H A198833 Vincenzo Librandi, <a href="/A198833/b198833.txt">Table of n, a(n) for n = 1..1000</a>
%H A198833 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1)
%F A198833 a(n) = n*(n+1)*(n^2+n+2)*(n^2+n+4)/48.
%F A198833 G.f.: x*(1+3*x+7*x^2+3*x^3+x^4) / (1-x)^7. - _R. J. Mathar_, Oct 30 2011
%F A198833 a(n) = Sum_{i=1..A000217(n)} A000217(i). [_Bruno Berselli_, Sep 06 2013]
%F A198833 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.
%F A198833 a(n) = A047780(n) - A093566(n+1) = (A047780(n) + A337898(n)) / 2 = A093566(n+1) + A337898(n). - _Robert A. Russell_, Oct 19 2020
%t A198833 Table[(n^6 + 3 n^5 + 9 n^4 + 13 n^3 + 14 n^2 + 8 n)/48, {n, 25}]
%t A198833 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 *)
%t A198833 LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,10,56,220,680,1771,4060},40] (* _Harvey P. Dale_, Nov 06 2024 *)
%o A198833 (PARI) a(n)=n*(n+1)*(n^2+n+2)*(n^2+n+4)/48 \\ _Charles R Greathouse IV_, Aug 02 2013
%o A198833 (Magma) [n*(n+1)*(n^2+n+2)*(n^2+n+4)/48: n in [1..35]]; // _Vincenzo Librandi_, Aug 04 2013
%Y A198833 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).
%Y A198833 Row 3 of A325005 (orthotope facets, orthoplex vertices) and A337888 (orthotope faces, orthoplex peaks).
%K A198833 nonn,easy,changed
%O A198833 1,2
%A A198833 _Geoffrey Critzer_, Oct 30 2011

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