A376747 -id:A376747 - cp's OEIS frontend

A376822 Number of colorings of a toroidal n X n grid using exactly two colors under translational symmetry.

0, 5, 62, 4154, 1342206, 1908897150, 11488774559742, 288230376353050814, 29850020237398264483838, 12676506002282327791964489726, 21970710674130840874443091905462270, 154866286100907105149651981766316633972734, 4427744605404865645682169434028029029963535286270

Offset: 1

Andrew Howroyd, Oct 05 2024

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50

Main diagonal of A294684.

Cf. A179043, A376747 (colors permutable), A376823, A376824, A376825.

a(n) = A179043(n) - 2.

A376748 Number of non-isomorphic colorings of a toroidal n X n grid using exactly three swappable colors.

Original entry on oeis.org

0, 3, 345, 447156, 5647919665, 694881637942816, 813943290958393433377, 8941884948534360647405572800, 912400181570021638669407666368774097, 858962534553352212055863239761275173880606456, 7425662396340624836407113113710889289196975262054947345, 587417576454184723055270940786413231085263155884260701824558793960

Offset: 1

Marko Riedel, Oct 03 2024

nonn

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

Marko Riedel et al., Burnside lemma and translational symmetries of the torus.
Marko Riedel, Maple code for sequence.

Main diagonal of A294792.

Cf. A376747, A376749, A376808, A376823.

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.

A376749 Number of non-isomorphic colorings of a toroidal n X n grid using exactly four swappable colors.

Original entry on oeis.org

0, 1, 874, 10741819, 1870851589562, 5465007068038102643, 269482732023591671431784330, 221537990355601030571170905795094315, 3007205014171762201565124875608675533096268906, 669557518440386985607930852942771727146772232484581602227, 2433673642945425535196140161775877796522974318753784273286700783313050

Offset: 1

Marko Riedel, Oct 03 2024

nonn

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

Marko Riedel et al., Burnside lemma and translational symmetries of the torus.
Marko Riedel, Maple code for sequence.

Main diagonal of A294793.

Cf. A376747, A376748, A376808, A376824.

a(n) = (1/(n^2*4!))*(Sum_{sigma in S_4} 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..4} (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.

cp's OEIS Frontend

A376822 Number of colorings of a toroidal n X n grid using exactly two colors under translational symmetry.

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A376748 Number of non-isomorphic colorings of a toroidal n X n grid using exactly three swappable colors.

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A376749 Number of non-isomorphic colorings of a toroidal n X n grid using exactly four swappable colors.

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