A321244 Non-isomorphic proper colorings of the 3 X 3 grid graph using at most n colors under rotational and reflectional symmetries.
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
References
- F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973.
Links
- Colin Barker, Table of n, a(n) for n = 1..1000
- 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.
- Marko Riedel, Maple code for OCP computation by Burnside.
- Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
Programs
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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
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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 *) -
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
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)