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A222187 Number of toroidal n X 2 binary arrays, allowing rotation and/or reflection of the rows and/or the columns. 1
3, 7, 13, 34, 78, 237, 687, 2299, 7685, 27190, 96909, 353384, 1296858, 4808707, 17920860, 67169299, 252745368, 954677597, 3617214681, 13744852240, 52359294790, 199915018057, 764884036743, 2932046213314, 11259024569838, 43303903226962, 166800088109829 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Table of n, a(n) for n=1..27.

S. N. Ethier, Counting toroidal binary arrays, arXiv preprint arXiv:1301.2352 [math.CO], 2013.

S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.

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)/(4n), (2^(m*n/2) + 2^((m+2)*n/2))/(8n)];

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

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}]/(8n)];

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[m_] := a[m, 2];

Array[a, 27] (* Jean-Fran├žois Alcover, Sep 23 2018, after Michel Marcus in A222188 *)

CROSSREFS

A column of A222188.

Sequence in context: A222227 A152981 A112040 * A084611 A078454 A023212

Adjacent sequences:  A222184 A222185 A222186 * A222188 A222189 A222190

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 11 2013

EXTENSIONS

More terms from Michel Marcus, Feb 17 2013

STATUS

approved

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Last modified October 21 11:42 EDT 2019. Contains 328296 sequences. (Running on oeis4.)