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A376748
Number of non-isomorphic colorings of a toroidal n X n grid using exactly three swappable colors.
4
0, 3, 345, 447156, 5647919665, 694881637942816, 813943290958393433377, 8941884948534360647405572800, 912400181570021638669407666368774097, 858962534553352212055863239761275173880606456, 7425662396340624836407113113710889289196975262054947345, 587417576454184723055270940786413231085263155884260701824558793960
OFFSET
1,2
REFERENCES
F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.
FORMULA
a(n) = (1/(n^2*3!))*(Sum_{sigma in S_3} Sum_{d|n} Sum_{f|n} phi(d) phi(f) [[forall j_l(sigma) > 0 : l|lcm(d,f) ]] P(gcd(d,f)*(n/d)*(n/f), sigma)) where P(F, sigma) = F! [z^F] Product_{l=1..3} (exp(lz)-1)^j_l(sigma). The notation j_l(sigma) is from the Harary text and gives the number of cycles of length l in the permutation sigma. [[.]] is an Iverson bracket.
CROSSREFS
Main diagonal of A294792.
Sequence in context: A249799 A234607 A057121 * A039515 A209426 A037302
KEYWORD
nonn
AUTHOR
Marko Riedel, Oct 03 2024
STATUS
approved