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A255015
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Number of toroidal n X n binary arrays, allowing rotation of rows and/or columns as well as matrix transposition.
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3
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1, 2, 6, 44, 2209, 674384, 954623404, 5744406453840, 144115192471496836, 14925010120653819583840, 6338253001142965335834871200, 10985355337065423791175013899922368, 77433143050453552587418968170813573149024
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = (2*n^2)^{-1} Sum_{ c divides n } Sum_{ d divides n } phi(c)*phi(d)* 2^(n^2/lcm(c,d)) + (2*n)^{-1} Sum_{ d divides n } phi(d)*2^(n*(n + d - 2 *floor(d/2))/(2*d)), where phi is A000010.
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MATHEMATICA
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a[n_] := (2 n^2)^(-1) Sum[If[Mod[n, c] == 0, Sum[If[Mod[n, d] == 0, EulerPhi[c] EulerPhi[d] 2^(n^2/ LCM[c, d]), 0], {d, 1, n}], 0], {c, 1, n}] + (2 n)^(-1) Sum[If[Mod[n, d] == 0, EulerPhi[d] 2^(n (n + d - 2 IntegerPart[d/2])/(2 d)), 0], {d, 1, n}];
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CROSSREFS
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Cf. A184271 (number of m X n binary arrays allowing rotation of rows/columns), A179043 (main diagonal of A184271), A222188 (number of m X n binary arrays allowing rotation/reflection of rows/columns), A209251 (main diagonal of A222188), A255016 (number of n X n binary arrays allowing rotation/reflection of rows/columns as well as matrix transposition).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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