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

Author

Marko Riedel, Nov 01 2018

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A002817, A321245, A321246.

Programs

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

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)