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A321244
Non-isomorphic proper colorings of the 3 X 3 grid graph using at most n colors under rotational and reflectional symmetries.
3
0, 2, 69, 1572, 19865, 153480, 830802, 3476144, 12003462, 35757630, 94780235, 228579252, 509929719, 1065625652, 2106541920, 3969848640, 7176749852, 12509692794, 21113614017, 34626453860, 55344881445, 86431928352, 132174030494, 198295824432, 292341936450, 424135940150, 606327641127, 855040875444, 1190635082147, 1638595028940
OFFSET
1,2
REFERENCES
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973.
LINKS
Marko Riedel et al., Tree graphs colorings, Math StackExchange, December 2017.
Marko Riedel et al., 3-colourings of a 3×3 table with one of 3 colors up to symmetries, Math StackExchange, October 2018.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
FORMULA
a(n) = (1/8)*n^9 - (3/2)*n^8 + (33/4)*n^7 - (53/2)*n^6 + (217/4)*n^5 - (291/4)*n^4 + (507/8)*n^3 - (133/4)*n^2 + 8*n.
From Colin Barker, Nov 01 2018: (Start)
G.f.: x^2*(2 + 49*x + 972*x^2 + 7010*x^3 + 17710*x^4 + 15273*x^5 + 4076*x^6 + 268*x^7) / (1 - x)^10.
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n>10.
(End)
MATHEMATICA
CoefficientList[Series[x (2 + 49 x + 972 x^2 + 7010 x^3 + 17710 x^4 + 15273 x^5 + 4076 x^6 + 268 x^7) / (1 - x)^10, {x, 0, 30}], x] (* Vincenzo Librandi Nov 04 2018 *)
PROG
(PARI) concat(0, Vec(x^2*(2 + 49*x + 972*x^2 + 7010*x^3 + 17710*x^4 + 15273*x^5 + 4076*x^6 + 268*x^7) / (1 - x)^10 + O(x^30))) \\ Colin Barker, Nov 01 2018
(Magma) [(1/8)*n^9-(3/2)*n^8+(33/4)*n^7-(53/2)*n^6+(217/4)*n^5-(291/4)*n^4 +(507/8)*n^3-(133/4)*n^2+8*n: n in [1..30]]; // Vincenzo Librandi, Nov 04 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Marko Riedel, Nov 01 2018
STATUS
approved