A209251 Number of n X n checkered tori, allowing rotation and/or reflection of the rows and/or the columns.

1, 2, 7, 36, 1459, 340880, 478070832, 2872221202512, 72057630729710704, 7462505061854009276768, 3169126500599982009308551168, 5492677668532714149024993226980288, 38716571525226776692749451887896112574464

Offset: 0

Author

Jonathan Vos Post, Jan 14 2013

Keywords

• nonn

Comments

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).

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50
• S. N. Ethier, Counting toroidal binary arrays, arXiv:1301.2352v1 [math.CO], Jan 10, 2013 and J. Int. Seq. 16 (2013) #13.4.7.
• S. N. Ethier and Jiyeon Lee, Counting toroidal binary arrays, II, arXiv:1502.03792v1 [math.CO], Feb 12, 2015 and J. Int. Seq. 18 (2015) # 15.8.3.

Crossrefs

Main diagonal of A222188.

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

Programs

Mathematica

b1[m_, n_] := Sum[EulerPhi[c]*EulerPhi[d]*2^(m*n/LCM[c, d]), {c, Divisors[m]}, {d, Divisors[n]}]/(4*m*n);
b2a[m_, n_] := If[OddQ[m], 2^((m + 1)*n/2)/(4*n), (2^(m*n/2) + 2^((m + 2)*n/2))/(8*n)];
b2b[m_, n_] := DivisorSum[n, If[# >= 2, EulerPhi[#]*2^((m*n)/#), 0] &]/(4*n);
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)];
b2[m_, n_] := b2a[m, n] + b2b[m, n] + b2c[m, n];
b3[m_, n_] := b2[n, m]; b4oo[m_, n_] := 2^((m*n - 3)/2);
b4eo[m_, n_] := 3*2^(m*n/2 - 3); b4ee[m_, n_] := 7*2^(m*n/2 - 4);
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]];
a[0] = 1; a[n_] := a[n, n];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Oct 08 2017, after Michel Marcus's code for A222188 *)

Extensions

More terms from Michel Marcus, Feb 13 2013

a(0)=1 prepended by Andrew Howroyd, Sep 30 2017