A222188 Table read by antidiagonals: number of toroidal m X n binary arrays, allowing rotation and/or reflection of the rows and/or the columns.
2, 3, 3, 4, 7, 4, 6, 13, 13, 6, 8, 34, 36, 34, 8, 13, 78, 158, 158, 78, 13, 18, 237, 708, 1459, 708, 237, 18, 30, 687, 4236, 14676, 14676, 4236, 687, 30, 46, 2299, 26412, 184854, 340880, 184854, 26412, 2299, 46
Offset: 1
Examples
Array begins: 2, 3, 4, 6, 8, 13, 18, 30, ... 3, 7, 13, 34, 78, 237, 687, 2299, ... 4, 13, 36, 158, 708, 4236, 26412, 180070, ... 6, 34, 158, 1459, 14676, 184854, 2445918, 33888844, ... 8, 78, 708, 14676, 340880, 8999762, 245619576, 6873769668, ... ...
Links
- G. C. Greubel, Table of n, a(n) for the first 100 antidiagonals, flattened
- S. N. Ethier, Counting toroidal binary arrays, arXiv:1301.2352 [math.CO], 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.
- S. N. Ethier and Jiyeon Lee, Parrondo games with two-dimensional spatial dependence, arXiv:1510.06947 [math.PR], 2015.
- Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023. See p. 3.
Crossrefs
Programs
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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]]; Table[a[m - n+1, n], {m, 1, 10}, {n, 1, m}] // Flatten (* Jean-François Alcover, Dec 05 2015, adapted from Michel Marcus's PARI script *) -
PARI
odd(n) = n%2; b1(m,n) = sumdiv(m, c, sumdiv(n, d, eulerphi(c)*eulerphi(d)*2^(m*n/lcm(c,d))))/(4*m*n); b2a(m,n) = if (odd(m), 2^((m+1)*n/2)/(4*n), (2^(m*n/2)+2^((m+2)*n/2))/(8*n)); b2b(m,n) = sumdiv(n, d, if (d>=2, eulerphi(d)*2^((m*n)/d), 0))/(4*n); b2c(m,n) = if (odd(m), sum(j=1, n-1, if (odd(n/gcd(j,n)), 2^((m+1)*gcd(j,n)/2)-2^(m*gcd(j,n))))/(4*n), sum(j=1, n-1, if (odd(n/gcd(j,n)), 2^(m*gcd(j,n)/2)+2^((m+2)*gcd(j,n)/2)-2^(m*gcd(j,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) = {if (odd(m), if (odd(n), b = b4oo(m,n), b = b4eo(m,n)), if (odd(n), b = b4eo(m,n), b = b4ee(m,n))); b += b1(m,n) + b2(m,n) + b3(m,n); return (b);} \\ Michel Marcus, Feb 13 2013