A294685 - OEIS

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.

0, 0, 9, 2, 91, 2022, 9, 738, 43315, 2679246, 30, 5613, 950062, 174184755, 33887517990, 91, 43404, 21480921, 11765865678, 6862930841141, 4169289730628814, 258, 338259, 497812638, 816999710223, 1429469771994078, 2605213713043722909, 4883659745750360600262, 729, 2679228, 11765822365, 57906482267826, 303941554100145501

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OFFSET

1,3

COMMENTS

Colors are not being permuted, i.e., Power Group Enumeration does not apply here.

REFERENCES

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..820 (first 40 rows)

Marko Riedel et al., Burnside lemma and translational symmetries of the torus.

FORMULA

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.

T(n,k) = A184284(n,k) - 3*A184271(n,k) + 3. - Andrew Howroyd, Oct 05 2024

EXAMPLE

Triangle begins:

0, 9;

2, 91, 2022;

9, 738, 43315, 2679246;

30, 5613, 950062, 174184755, 33887517990;

91, 43404, 21480921, 11765865678, 6862930841141, 4169289730628814;

...

PROG

(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

CROSSREFS

Main diagonal is A376823.

Cf. A184271, A184284, A294684, A294686, A294687, A294791, A294792, A294793, A294794. T(n,1) is A056283.

Sequence in context: A174837 A096042 A038292 * A200238 A368982 A254666

Adjacent sequences: A294682 A294683 A294684 * A294686 A294687 A294688

KEYWORD

nonn,tabl,nice

AUTHOR

Marko Riedel, Nov 06 2017

STATUS

approved