A184271 -id:A184271 - cp's OEIS frontend

A184291 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..5 arrays.

6, 21, 21, 76, 351, 76, 336, 7826, 7826, 336, 1560, 210456, 1119936, 210456, 1560, 7826, 6047412, 181402676, 181402676, 6047412, 7826, 39996, 181410426, 31345666736, 176319685116, 31345666736, 181410426, 39996, 210126, 5597460306

Offset: 1

R. H. Hardin, Jan 10 2011

			Table starts
      6        21          76          336        1560          7826      39996
     21       351        7826       210456     6047412     181410426 5597460306
     76      7826     1119936    181402676 31345666736 5642220395616
    336    210456   181402676 176319685116
   1560   6047412 31345666736
   7826 181410426
  39996

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 31 terms from R. H. Hardin)
S. N. Ethier, Counting toroidal binary arrays, arXiv:1301.2352v1 [math.CO], Jan 10, 2013.
S. N. Ethier and Jiyeon Lee, Counting toroidal binary arrays, II, arXiv:1502.03792v1 [math.CO], Feb 12, 2015.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.

Columns 1-3 are A054625, A184289, A184290.

Cf. A184271, A184284.

T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*6^(n*(k/LCM[c, d])), {d, Divisors[k]}], {c, Divisors[n]}]; Table[T[n-k+1, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 30 2017, after Andrew Howroyd *)

T(n, k) = (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 6^(n*k/lcm(c,d)))); \\ Andrew Howroyd, Sep 27 2017

T(n,k) = (1/(n*k)) * Sum_{c|n} Sum_{d|k} phi(c) * phi(d) * 6^(n*k/lcm(c,d)). - Andrew Howroyd, Sep 27 2017

A184331 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..6 arrays.

Original entry on oeis.org

7, 28, 28, 119, 637, 119, 616, 19684, 19684, 616, 3367, 721525, 4484039, 721525, 3367, 19684, 28249228, 1153450872, 1153450872, 28249228, 19684, 117655, 1153470437, 316504102999, 2077059243301, 316504102999, 1153470437, 117655, 720916

Offset: 1

R. H. Hardin, Jan 11 2011

			Table starts
       7         28          119           616         3367          19684
      28        637        19684        721525     28249228     1153470437
     119      19684      4484039    1153450872 316504102999 90467424400444
     616     721525   1153450872 2077059243301
    3367   28249228 316504102999
   19684 1153470437
  117655

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 31 terms from R. H. Hardin)
S. N. Ethier, Counting toroidal binary arrays, arXiv:1301.2352v1 [math.CO], Jan 10, 2013.
S. N. Ethier and Jiyeon Lee, Counting toroidal binary arrays, II, arXiv:1502.03792v1 [math.CO], Feb 12, 2015.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.

Columns 1-3 are A054626, A184329, A184330.

Cf. A184271, A184284.

T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*7^(n*(k/LCM[c, d])), {d, Divisors[k]}], {c, Divisors[n]}]; Table[T[n - k + 1, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 30 2017, after Andrew Howroyd *)

T(n, k) = (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 7^(n*k/lcm(c,d)))); \\ Andrew Howroyd, Sep 27 2017

T(n,k) = (1/(n*k)) * Sum_{c|n} Sum_{d|k} phi(c) * phi(d) * 7^(n*k/lcm(c,d)). - Andrew Howroyd, Sep 27 2017

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

A368306 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal reflections by a tile that is not fixed under horizontal reflection.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 4, 8, 9, 4, 4, 24, 32, 26, 4, 8, 56, 186, 182, 62, 9, 10, 190, 1096, 2130, 1096, 205, 10, 20, 596, 7356, 26296, 26380, 7356, 623, 22, 30, 2102, 49940, 350316, 671104, 350584, 49940, 2171, 30

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     24       56       190
   3 | 2   9   32    186     1096      7356
   4 | 4  26  182   2130    26296    350316
   5 | 4  62 1096  26380   671104  17899020
   6 | 9 205 7356 350584 17897924 954481360

Peter Kagey, Illustration of T(3,3)=32
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Cf. A368222, A368259, A368304, A368305, A368308, A184271.

A368306[n_, m_] := 1/(2*n*m)*(DivisorSum[n, Function[d, DivisorSum[m, EulerPhi[#] EulerPhi[d] 2^(m*n/LCM[#, d]) &]]] + n*If[EvenQ[n], DivisorSum[m, EulerPhi[#] (2^(n*m/LCM[2, #]) + 2^((n - 2)*m/LCM[2, #])*4^(m/#)*Boole[EvenQ[#]]) &]/2, DivisorSum[m, EulerPhi[#]*2^(n*m/#) &, EvenQ]])

A368308 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to 180-degree rotation by a tile that is not fixed under 180-degree rotation.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 4, 9, 9, 4, 4, 26, 32, 26, 4, 9, 62, 192, 192, 62, 9, 10, 205, 1096, 2174, 1096, 205, 10, 22, 623, 7440, 26500, 26500, 7440, 623, 22, 30, 2171, 49940, 351336, 671104, 351336, 49940, 2171, 30

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   9   32    192     1096      7440
   4 | 4  26  192   2174    26500    351336
   5 | 4  62 1096  26500   671104  17904476
   6 | 9 205 7440 351336 17904476 954546880

Peter Kagey, Illustration of T(3,3)=32
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Cf. A184271, A368224, A368263, A368304, A368306, A368307.

A368308[n_, m_] := 1/(2*n*m)*(DivisorSum[n, Function[d, DivisorSum[m, EulerPhi[#] EulerPhi[d] 2^(m*n/LCM[#, d]) &]]] + n*m*2^(n*m/2)*Which[OddQ[n*m], 0, OddQ[n + m], 1/2, True, 3/4])

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

A184264 Number of distinct n X 2 toroidal binary arrays.

Original entry on oeis.org

3, 7, 14, 40, 108, 362, 1182, 4150, 14602, 52588, 190746, 699600, 2581428, 9588742, 35792568, 134223910, 505294128, 1908896442, 7233642930, 27487869472, 104715443852, 399822696082, 1529755490574, 5864063066500, 22517998808028, 86607689013412, 333599974893066

Offset: 1

R. H. Hardin, Jan 10 2011

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000
S. N. Ethier, Counting toroidal binary arrays, arXiv preprint arXiv:1301.2352, 2013 and J. Int. Seq. 16 (2013) #13.4.7 .

Column 2 of A184271.

with(numtheory):
a:= n-> add(add(phi(c)*phi(d) *2^(2*n/ilcm(c, d)),
        d=divisors(n)), c=[1,2])/(2*n):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 25 2012

a(n) ~ 2^(2*n-1) / n. - Vaclav Kotesovec, Sep 04 2014

More terms from Alois P. Heinz, Aug 25 2012

A184265 Number of distinct n X 3 toroidal binary arrays.

Original entry on oeis.org

4, 14, 64, 352, 2192, 14624, 99880, 699252, 4971184, 35792568, 260301176, 1908882592, 14096303344, 104715443852, 781874941184, 5864062367252, 44152937528384, 333599974922264, 2528336632928152, 19215358428046176, 146402730743992960, 1117984489446008100

Offset: 1

R. H. Hardin, Jan 10 2011

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
S. N. Ethier, Counting toroidal binary arrays, arXiv preprint arXiv:1301.2352, 2013 and J. Int. Seq. 16 (2013) #13.4.7 .

Column 3 of A184271.

with(numtheory):
a:= n-> add(add(phi(c)*phi(d) *2^(3*n/ilcm(c, d)),
        d=divisors(n)), c=[1, 3])/(3*n):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 25 2012

a(n) ~ 8^n / (3*n). - Vaclav Kotesovec, Sep 04 2014

More terms from Alois P. Heinz, Aug 25 2012

A184266 Number of distinct n X 4 toroidal binary arrays.

Original entry on oeis.org

6, 40, 352, 4156, 52488, 699600, 9587580, 134223976, 1908881900, 27487869472, 399822505524, 5864063067176, 86607686432616, 1286742765058960, 19215358428046176, 288230376353050696, 4340410370537249376, 65588423376053309360, 994182417449857925988

Offset: 1

R. H. Hardin, Jan 10 2011

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..820
S. N. Ethier, Counting toroidal binary arrays, arXiv preprint arXiv:1301.2352, 2013 and J. Int. Seq. 16 (2013) #13.4.7.

Column 4 of A184271.

with(numtheory):
a:= n-> add(add(phi(c)*phi(d) *2^(4*n/ilcm(c, d)),
        d=divisors(n)), c=[1,2,4])/(4*n):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 25 2012

a(n) ~ 2^(4*n-2) / n. - Vaclav Kotesovec, Sep 04 2014

More terms from Alois P. Heinz, Aug 25 2012

A184267 Number of distinct n X 5 toroidal binary arrays.

Original entry on oeis.org

8, 108, 2192, 52488, 1342208, 35792568, 981706832, 27487816992, 781874936816, 22517998808448, 655069036708592, 19215358428046176, 567592125344909792, 16865594582168158776, 503719091506096394752, 15111572745196608608736, 455125014443154512836736

Offset: 1

R. H. Hardin, Jan 10 2011

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..660
S. N. Ethier, Counting toroidal binary arrays, arXiv preprint arXiv:1301.2352, 2013 and J. Int. Seq. 16 (2013) #13.4.7 .

Column 5 of A184271.

with(numtheory):
a:= n-> add(add(phi(c)*phi(d) *2^(5*n/ilcm(c, d)),
        d=divisors(n)), c=[1,5])/(5*n):
seq(a(n), n=1..25);  # Alois P. Heinz, Aug 25 2012

a(n) ~ 32^n / (5*n). - Vaclav Kotesovec, Sep 04 2014

More terms from Alois P. Heinz, Aug 25 2012

cp's OEIS Frontend

A184291 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..5 arrays.

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A184331 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..6 arrays.

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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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A368306 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal reflections by a tile that is not fixed under horizontal reflection.

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A368308 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to 180-degree rotation by a tile that is not fixed under 180-degree rotation.

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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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A184264 Number of distinct n X 2 toroidal binary arrays.

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A184265 Number of distinct n X 3 toroidal binary arrays.

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