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Number of n X n checkered tori, allowing rotation and/or reflection of the rows and/or the columns.
4

%I #33 Jun 09 2018 08:43:20

%S 1,2,7,36,1459,340880,478070832,2872221202512,72057630729710704,

%T 7462505061854009276768,3169126500599982009308551168,

%U 5492677668532714149024993226980288,38716571525226776692749451887896112574464

%N Number of n X n checkered tori, allowing rotation and/or reflection of the rows and/or the columns.

%C Main diagonal from p. 8, Ethier, of Table 4: The number b(m, n) of toroidal m X n binary arrays, allowing rotation and/or reflection of the rows and/or the columns, for m, n = 1, 2, ..., 8 (cf. A222188).

%H Andrew Howroyd, <a href="/A209251/b209251.txt">Table of n, a(n) for n = 0..50</a>

%H S. N. Ethier, <a href="http://arxiv.org/abs/1301.2352">Counting toroidal binary arrays</a>, arXiv:1301.2352v1 [math.CO], Jan 10, 2013 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Ethier/ethier2.html">J. Int. Seq. 16 (2013) #13.4.7</a>.

%H S. N. Ethier and Jiyeon Lee, <a href="http://arxiv.org/abs/1502.03792">Counting toroidal binary arrays, II</a>, arXiv:1502.03792v1 [math.CO], Feb 12, 2015 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Lee/lee6.html">J. Int. Seq. 18 (2015) # 15.8.3</a>.

%t b1[m_, n_] := Sum[EulerPhi[c]*EulerPhi[d]*2^(m*n/LCM[c, d]), {c, Divisors[m]}, {d, Divisors[n]}]/(4*m*n);

%t b2a[m_, n_] := If[OddQ[m], 2^((m + 1)*n/2)/(4*n), (2^(m*n/2) + 2^((m + 2)*n/2))/(8*n)];

%t b2b[m_, n_] := DivisorSum[n, If[# >= 2, EulerPhi[#]*2^((m*n)/#), 0] &]/(4*n);

%t b2c[m_, n_] := If[OddQ[m], Sum[If[OddQ[n/GCD[j, n]], 2^((m + 1)*GCD[j, n]/2) - 2^(m*GCD[j, n]), 0], {j, 1, n - 1}]/(4*n), Sum[If[OddQ[n/GCD[j, n]], 2^(m*GCD[j, n]/2) + 2^((m + 2)*GCD[j, n]/2) - 2^(m*GCD[j, n] + 1), 0], {j, 1, n - 1}]/(8*n)];

%t b2[m_, n_] := b2a[m, n] + b2b[m, n] + b2c[m, n];

%t b3[m_, n_] := b2[n, m]; b4oo[m_, n_] := 2^((m*n - 3)/2);

%t b4eo[m_, n_] := 3*2^(m*n/2 - 3); b4ee[m_, n_] := 7*2^(m*n/2 - 4);

%t a[m_, n_] := Module[{b}, If[OddQ[m], If[OddQ[n], b = b4oo[m, n], b = b4eo[m, n]], If[OddQ[n], b = b4eo[m, n], b = b4ee[m, n]]]; b += b1[m, n] + b2[m, n] + b3[m, n]; Return[b]];

%t a[0] = 1; a[n_] := a[n, n];

%t Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Oct 08 2017, after _Michel Marcus_'s code for A222188 *)

%Y Main diagonal of A222188.

%Y Cf. A179043, A184271 (n X k toroidal binary arrays).

%K nonn

%O 0,2

%A _Jonathan Vos Post_, Jan 14 2013

%E More terms from _Michel Marcus_, Feb 13 2013

%E a(0)=1 prepended by _Andrew Howroyd_, Sep 30 2017