A222188 -id:A222188 - cp's OEIS frontend

A255016 Number of toroidal n X n binary arrays, allowing rotation and/or reflection of rows and/or columns as well as matrix transposition.

1, 2, 6, 26, 805, 172112, 239123150, 1436120190288, 36028817512382026, 3731252531904348833632, 1584563250300891724601560272, 2746338834266358751489231123956672, 19358285762613388352671214587818634041520

Offset: 0

Jiyeon Lee, Feb 12 2015

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..57
S. N. Ethier and Jiyeon Lee, Counting toroidal binary arrays, II, arXiv:1502.03792 [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 preprint 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.
Wikipedia, Pólya enumeration theorem.

Cf. A184271 (number of m X n binary arrays allowing rotation of rows/columns), A179043 (main diagonal of A184271), A222188 (number of m X n binary arrays allowing rotation/reflection of rows/columns), A209251 (main diagonal of A222188), A255015 (number of n X n binary arrays allowing rotation of rows/columns as well as matrix transposition).

Cf. A054247.

a[n_] := (8 n^2)^(-1) Sum[If[Mod[n, c] == 0, Sum[If[Mod[n, d] == 0, EulerPhi[c] EulerPhi[d] 2^(n^2/ LCM[c, d]), 0], {d, 1, n}], 0], {c, 1, n}] + (4 n)^(-1) Sum[If[Mod[n, d] == 0, EulerPhi[d] 2^(n^2/d), 0], {d, 1, n}] + If[Mod[n, 2] == 1, (4 n)^(-1) Sum[If[Mod[n, d] == 0 && Mod[d, 2] == 1, EulerPhi[d] (2^(n (n + 1)/(2 d)) - 2^(n^2/d)), 0], {d, 1, n}],(8 n)^(-1) Sum[If[Mod[n, d] == 0 && Mod[d, 2] == 1, EulerPhi[d] (2^(n^2/(2 d)) + 2^(n (n + 2)/(2 d)) - 2 2^(n^2/d)), 0], {d, 1, n}]] + (1/2) If[Mod[n, 2] == 1, 2^((n^2 - 3)/2), 7 2^(n^2/2 - 4)] + (4 n)^(-1) Sum[If[Mod[n, d] == 0, EulerPhi[d] 2^(n (n + d - 2 IntegerPart[d/2])/(2 d)), 0], {d, 1, n}] + If[Mod[n, 2] == 1, 2^((n^2 - 5)/4), 5 2^(n^2/4 - 3)];

a(0)=1 from Alois P. Heinz, Feb 19 2015

A368304 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal and vertical reflections by an asymmetric tile.

Original entry on oeis.org

1, 4, 4, 6, 28, 6, 23, 194, 194, 23, 52, 2196, 7296, 2196, 52, 194, 26524, 350573, 350573, 26524, 194, 586, 351588, 17895736, 67136624, 17895736, 351588, 586, 2131, 4798174, 954495904, 13744131446, 13744131446, 954495904, 4798174, 2131

Offset: 1

Peter Kagey, Dec 21 2023

			Table begins:
  n\k|   1      2         3             4                5
  ---+----------------------------------------------------
   1 |   1      4         6            23               52
   2 |   4     28       194          2196            26524
   3 |   6    194      7296        350573         17895736
   4 |  23   2196    350573      67136624      13744131446
   5 |  52  26524  17895736   13744131446   11258999068672
   6 | 194 351588 954495904 2932037300956 9607679419823148

Peter Kagey, Illustration of T(2,2)=28
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Cf. A222188, A368220, A368257, A368302, A368303, A368306, A368308, A184271.

A368304[n_,m_]:=1/(4*n*m) (DivisorSum[n, Function[d,DivisorSum[m,Function[c,EulerPhi[c]EulerPhi[d]4^(m*n/LCM[c,d])]]]]+If[EvenQ[n],n/2*DivisorSum[m, EulerPhi[#](4^(n*m/LCM[2,#])+4^((n-2)*m/LCM[2,#])*4^(2m/#)*Boole[EvenQ[#]])&],n*DivisorSum[m,EulerPhi[#](4^(n*m/#))&,EvenQ]]+If[EvenQ[m], m/2*DivisorSum[n,EulerPhi[#](4^(n*m/LCM[2,#])+4^((m-2)*n/LCM[2,#])*4^(2n/#)*Boole[EvenQ[#]])&],m*DivisorSum[n, EulerPhi[#](4^(m*n/#))&,EvenQ]]+Which[EvenQ[n]&&EvenQ[m],(n*m)/4 (3*2^(n*m)),OddQ[n*m],0,OddQ[n+m],(n*m)/2 (2^(n*m))])

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

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Jan 14 2013

nonn

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).

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.

Main diagonal of A222188.

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

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 *)

More terms from Michel Marcus, Feb 13 2013

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

A368302 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal and vertical reflections by a tile that is fixed under horizontal reflections but not vertical reflections.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 4, 9, 8, 4, 4, 26, 22, 22, 4, 9, 62, 120, 126, 44, 8, 10, 205, 600, 1267, 592, 135, 9, 22, 623, 3936, 14164, 13600, 3936, 362, 18, 30, 2171, 25556, 181782, 337192, 178366, 25314, 1211, 23, 62, 7429, 177678, 2437726, 8965354, 8980642, 2404372, 176998, 3914, 44

Offset: 1

Peter Kagey, Dec 21 2023

			Table begins:
  n\k| 1   2    3      4       5         6
  ---+------------------------------------
   1 | 1   2    2      4       4         9
   2 | 2   5    9     26      62       205
   3 | 2   8   22    120     600      3936
   4 | 4  22  126   1267   14164    181782
   5 | 4  44  592  13600  337192   8965354
   6 | 8 135 3936 178366 8980642 477655760

Peter Kagey, Illustration of T(3,3)=22
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Cf. A222188, A368218, A368254, A368303, A368304, A368305.

A368302[n_, m_] := 1/(4*n*m) (DivisorSum[n, Function[d, DivisorSum[m, EulerPhi[#] EulerPhi[d] 2^(m*n/LCM[#, d]) &]]] + n*If[EvenQ[n], 1/2*DivisorSum[m, EulerPhi[#] (2^(n*m/LCM[2, #]) + 2^((n - 2)*m/LCM[2, #])*2^(2 m/#)) &], DivisorSum[m, EulerPhi[#] (2^((n - 1)*m/LCM[2, #])*2^(m/#)) &]] + m*If[EvenQ[m], 1/2*DivisorSum[n, EulerPhi[#] (2^(n*m/LCM[2, #]) + 2^(m*n/#)*Boole[EvenQ[#]]) &], DivisorSum[n, EulerPhi[#]*2^(m*n/#) &, EvenQ]] + n*m*2^(n*m/2)*Which[EvenQ[n] && EvenQ[m], 3/4, OddQ[n*m], 0, OddQ[n + m], 1/2])

A368303 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal and vertical reflections by a tile that is fixed under 180-degree rotations but not horizontal or vertical reflections.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 4, 8, 8, 4, 4, 22, 24, 22, 4, 8, 44, 120, 120, 44, 8, 9, 135, 612, 1203, 612, 135, 9, 18, 362, 3892, 13600, 13600, 3892, 362, 18, 23, 1211, 25482, 177342, 337600, 177342, 25482, 1211, 23, 44, 3914, 176654, 2404372, 8962618, 8962618, 2404372, 176654, 3914, 44

Offset: 1

Peter Kagey, Dec 21 2023

			Table begins:
  n\k| 1   2    3      4       5         6
  ---+------------------------------------
   1 | 1   2    2      4       4         8
   2 | 2   5    8     22      44       135
   3 | 2   8   24    120     612      3892
   4 | 4  22  120   1203   13600    177342
   5 | 4  44  612  13600  337600   8962618
   6 | 8 135 3892 177342 8962618 477371760

Peter Kagey, Illustration of T(3,3)=24
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Cf. A222188, A368219, A368256, A368302, A368304, A368307.

A368303[n_, m_]:=1/(4*n*m)*(DivisorSum[n, Function[d, DivisorSum[m, Function[c, EulerPhi[c]EulerPhi[d]2^(m*n/LCM[c, d])]]]] + If[EvenQ[n], n/2*DivisorSum[m, EulerPhi[#](2^(n*m/LCM[2, #]) + 2^((n - 2)*m/LCM[2, #])*2^(2m/#)*Boole[EvenQ[#]])&], n*DivisorSum[m, EulerPhi[#](2^(n*m/#))&, EvenQ]] + If[EvenQ[m], m/2*DivisorSum[n, EulerPhi[#](2^(n*m/LCM[2, #]) + 2^((m - 2)*n/LCM[2, #])*2^(2n/#)*Boole[EvenQ[#]])&], m*DivisorSum[n, EulerPhi[#](2^(m*n/#))&, EvenQ]] + n*m*2^((n*m)/2)*Which[OddQ[n*m], Sqrt[2], OddQ[n + m], 3/2, True, 7/4])

A222187 Number of toroidal n X 2 binary arrays, allowing rotation and/or reflection of the rows and/or the columns.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Feb 11 2013

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..1667
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.
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.

A column of A222188.

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-François Alcover, Sep 23 2018, after Michel Marcus in A222188 *)

More terms from Michel Marcus, Feb 17 2013

A255015 Number of toroidal n X n binary arrays, allowing rotation of rows and/or columns as well as matrix transposition.

Original entry on oeis.org

1, 2, 6, 44, 2209, 674384, 954623404, 5744406453840, 144115192471496836, 14925010120653819583840, 6338253001142965335834871200, 10985355337065423791175013899922368, 77433143050453552587418968170813573149024

Offset: 0

Stewart N. Ethier, Feb 12 2015

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..57
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.
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.
Wikipedia, Polya enumeration theorem.

Cf. A184271 (number of m X n binary arrays allowing rotation of rows/columns), A179043 (main diagonal of A184271), A222188 (number of m X n binary arrays allowing rotation/reflection of rows/columns), A209251 (main diagonal of A222188), A255016 (number of n X n binary arrays allowing rotation/reflection of rows/columns as well as matrix transposition).

a[n_] := (2 n^2)^(-1) Sum[If[Mod[n, c] == 0, Sum[If[Mod[n, d] == 0, EulerPhi[c] EulerPhi[d] 2^(n^2/ LCM[c, d]), 0], {d, 1, n}], 0], {c, 1, n}] + (2 n)^(-1) Sum[If[Mod[n, d] == 0, EulerPhi[d] 2^(n (n + d - 2 IntegerPart[d/2])/(2 d)), 0], {d, 1, n}];

a(n) = (2*n^2)^{-1} Sum_{ c divides n } Sum_{ d divides n } phi(c)*phi(d)* 2^(n^2/lcm(c,d)) + (2*n)^{-1} Sum_{ d divides n } phi(d)*2^(n*(n + d - 2 *floor(d/2))/(2*d)), where phi is A000010.

a(0)=1 from Alois P. Heinz, Feb 19 2015

A368253 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal and vertical reflections by two tiles that are fixed under these reflections.

Original entry on oeis.org

2, 3, 3, 6, 7, 4, 10, 24, 13, 6, 20, 76, 74, 34, 8, 36, 288, 430, 378, 78, 13, 72, 1072, 3100, 4756, 1884, 237, 18, 136, 4224, 23052, 70536, 53764, 11912, 687, 30, 272, 16576, 179736, 1083664, 1689608, 709316, 77022, 2299, 46

Offset: 1

Peter Kagey, Dec 19 2023

			Table begins:
  n\k |  1   2     3      4        5          6
  ----+----------------------------------------
    1 |  2   3     6     10       20         36
    2 |  3   7    24     76      288       1072
    3 |  4  13    74    430     3100      23052
    4 |  6  34   378   4756    70536    1083664
    5 |  8  78  1884  53764  1689608   53762472
    6 | 13 237 11912 709316 44900448 2865540112

Peter Kagey, Illustration of T(2,3)=24
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023.

Cf. A222188, A225910.

Cf. A005418 (n=1), A225826 (n=2), A000029 (k=1), A222187 (k=2).

A368253[n_, m_] := 1/(4n)*(DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/d)]] + n*If[EvenQ[n], 1/2 (2^((n*m + 2 m)/2) + 2^(n*m/2)), 2^((n*m + m)/2)] + If[EvenQ[m], DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/LCM[d, 2])]], DivisorSum[n, Function[d, EulerPhi[d]*2^((n*m - n)/LCM[d, 2])*2^(n/d)]]] + n*Which[EvenQ[m], 2^(n*m/2), OddQ[m] && EvenQ[n], (3/2*2^(n*m/2)), OddQ[m] && OddQ[n], 2^((n*m + 1)/2)])

A222189 Number of toroidal n X 3 binary arrays, allowing rotation and/or reflection of the rows and/or the columns.

Original entry on oeis.org

4, 13, 36, 158, 708, 4236, 26412, 180070, 1256914, 8999762, 65225244, 477772294, 3525803320, 26185264801, 195490126328, 1466095545930, 11038514989344, 83401050695432, 632087998742988, 4803854169636124, 36600736833265244, 279496328812771427

Offset: 1

N. J. A. Sloane, Feb 12 2013

nonn

S. N. Ethier, Counting toroidal binary arrays, arXiv preprint arXiv:1301.2352, 2013 and J. Int. Seq. 16 (2013) #13.4.7.

A column of A222188.

More terms from Michel Marcus, Feb 13 2013

A222190 Number of toroidal n X 4 binary arrays, allowing rotation and/or reflection of the rows and/or the columns.

Original entry on oeis.org

6, 34, 158, 1459, 14676, 184854, 2445918, 33888844, 479268556, 6886509940, 100056170778, 1466749421254, 21657254505396, 321725243553514, 4804133557262316, 72059797641376804, 1085119161923382576, 16397231016301629254, 248546552679488856498

Offset: 1

N. J. A. Sloane, Feb 12 2013

nonn

S. N. Ethier, Counting toroidal binary arrays, arXiv preprint arXiv:1301.2352, 2013 and J. Int. Seq. 16 (2013) #13.4.7.

A column of A222188.

More terms from Michel Marcus, Feb 13 2013

cp's OEIS Frontend

A255016 Number of toroidal n X n binary arrays, allowing rotation and/or reflection of rows and/or columns as well as matrix transposition.

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A368304 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal and vertical reflections by an asymmetric tile.

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

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A368302 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal and vertical reflections by a tile that is fixed under horizontal reflections but not vertical reflections.

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A368303 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal and vertical reflections by a tile that is fixed under 180-degree rotations but not horizontal or vertical reflections.

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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.

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A255015 Number of toroidal n X n binary arrays, allowing rotation of rows and/or columns as well as matrix transposition.

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A368253 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal and vertical reflections by two tiles that are fixed under these reflections.

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A222189 Number of toroidal n X 3 binary arrays, allowing rotation and/or reflection of the rows and/or the columns.

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