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A294685 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly three colors under translational symmetry, 1 <= k <= n.

%I A294685 #37 Oct 05 2024 16:30:57
%S A294685 0,0,9,2,91,2022,9,738,43315,2679246,30,5613,950062,174184755,
%T A294685 33887517990,91,43404,21480921,11765865678,6862930841141,
%U A294685 4169289730628814,258,338259,497812638,816999710223,1429469771994078,2605213713043722909,4883659745750360600262,729,2679228,11765822365,57906482267826,303941554100145501
%N A294685 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly three colors under translational symmetry, 1 <= k <= n.
%C A294685 Colors are not being permuted, i.e., Power Group Enumeration does not apply here.
%D A294685 F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.
%H A294685 Andrew Howroyd, <a href="/A294685/b294685.txt">Table of n, a(n) for n = 1..820</a> (first 40 rows)
%H A294685 Marko Riedel et al., <a href="https://math.stackexchange.com/questions/2506511/">Burnside lemma and translational symmetries of the torus.</a>
%F A294685 T(n,k) = (Q!/(n*k))*(Sum_{d|n} Sum_{f|k} phi(d) phi(f) S(gcd(d,f)*(n/d)*(k/f), Q)) with Q=3 and S(n,k) Stirling numbers of the second kind.
%F A294685 T(n,k) = A184284(n,k) - 3*A184271(n,k) + 3. - _Andrew Howroyd_, Oct 05 2024
%e A294685 Triangle begins:
%e A294685    0;
%e A294685    0,     9;
%e A294685    2,    91,     2022;
%e A294685    9,   738,    43315,     2679246;
%e A294685   30,  5613,   950062,   174184755,   33887517990;
%e A294685   91, 43404, 21480921, 11765865678, 6862930841141, 4169289730628814;
%e A294685   ...
%o A294685 (PARI) T(n,m)=6*sumdiv(n, d, sumdiv(m, e, eulerphi(d) * eulerphi(e) * stirling(n*m/lcm(d,e), 3, 2) ))/(n*m) \\ _Andrew Howroyd_, Oct 05 2024
%Y A294685 Main diagonal is A376823.
%Y A294685 Cf. A184271, A184284, A294684, A294686, A294687, A294791, A294792, A294793, A294794. T(n,1) is A056283.
%K A294685 nonn,tabl,nice
%O A294685 1,3
%A A294685 _Marko Riedel_, Nov 06 2017