|
%I #20 Nov 30 2023 12:26:13
%S 3,7,13,34,78,237,687,2299,7685,27190,96909,353384,1296858,4808707,
%T 17920860,67169299,252745368,954677597,3617214681,13744852240,
%U 52359294790,199915018057,764884036743,2932046213314,11259024569838,43303903226962,166800088109829
%N Number of toroidal n X 2 binary arrays, allowing rotation and/or reflection of the rows and/or the columns.
%H Michael De Vlieger, <a href="/A222187/b222187.txt">Table of n, a(n) for n = 1..1667</a>
%H S. N. Ethier, <a href="https://arxiv.org/abs/1301.2352">Counting toroidal binary arrays</a>, arXiv preprint arXiv:1301.2352 [math.CO], 2013.
%H S. N. Ethier, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Ethier/ethier2.html">Counting toroidal binary arrays</a>, J. Int. Seq. 16 (2013) #13.4.7.
%H Peter Kagey and William Keehn, <a href="https://arxiv.org/abs/2311.13072">Counting tilings of the n X m grid, cylinder, and torus</a>, arXiv:2311.13072 [math.CO], 2023. See p. 3.
%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)/(4n), (2^(m*n/2) + 2^((m+2)*n/2))/(8n)];
%t b2b[m_, n_] := DivisorSum[n, If[# >= 2, EulerPhi[#]*2^((m*n)/#), 0]&]/(4n);
%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}]/(8n)];
%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[m_] := a[m, 2];
%t Array[a, 27] (* _Jean-François Alcover_, Sep 23 2018, after _Michel Marcus_ in A222188 *)
%Y A column of A222188.
%K nonn
%O 1,1
%A _N. J. A. Sloane_, Feb 11 2013
%E More terms from _Michel Marcus_, Feb 17 2013
|
