A179043 -id:A179043 - cp's OEIS frontend

A376822 Number of colorings of a toroidal n X n grid using exactly two colors under translational symmetry.

0, 5, 62, 4154, 1342206, 1908897150, 11488774559742, 288230376353050814, 29850020237398264483838, 12676506002282327791964489726, 21970710674130840874443091905462270, 154866286100907105149651981766316633972734, 4427744605404865645682169434028029029963535286270

Offset: 1

Andrew Howroyd, Oct 05 2024

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50

Main diagonal of A294684.

Cf. A179043, A376747 (colors permutable), A376823, A376824, A376825.

a(n) = A179043(n) - 2.

A376823 Number of colorings of a toroidal n X n grid using exactly three colors under translational symmetry.

Original entry on oeis.org

0, 9, 2022, 2679246, 33887517990, 4169289730628814, 4883659745750360600262, 53651309691205903304049168186, 5474401089420129832016444491921748358, 5153775207320113272335114827748860107542139918, 44553974378043749018442678682265335735181851572329684070

Offset: 1

Andrew Howroyd, Oct 05 2024

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..40

Main diagonal of A294685.

Cf. A179043, A184278, A376748 (colors permutable), A376822, A376824, A376825.

a(n) = A184278(n) - 3*A179043(n) + 3.

A376824 Number of colorings of a toroidal n X n grid using exactly four colors under translational symmetry.

Original entry on oeis.org

0, 6, 20720, 257706024, 44900438149488, 131160169581733489616, 6467585568566200114362823920, 5316911768534424725926923896066891424, 72172920340122292837562997014593985220649867760, 16069380442569287654590340470284256047904187412954757496784

Offset: 1

Andrew Howroyd, Oct 05 2024

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..40

Main diagonal of A294686.

Cf. A179043, A184272, A184278, A376749 (colors permutable), A376822, A376823, A376825.

a(n) = A184272(n) - 4*A184278(n) + 6*A179043(n) - 4.

A323862 Table read by antidiagonals where A(n,k) is the number of n X k binary arrays in which both the sequence of rows and the sequence of columns are (independently) aperiodic.

Original entry on oeis.org

2, 2, 2, 6, 10, 6, 12, 54, 54, 12, 30, 228, 498, 228, 30, 54, 990, 4020, 4020, 990, 54, 126, 3966, 32730, 65040, 32730, 3966, 126, 240, 16254, 261522, 1047540, 1047540, 261522, 16254, 240, 504, 65040, 2097018, 16768860, 33554370, 16768860, 2097018, 65040, 504

Offset: 1

Gus Wiseman, Feb 04 2019

A sequence of length n is aperiodic if all n rotations of its entries are distinct.

			Array begins:
        2        2        6       12       30
        2       10       54      228      990
        6       54      498     4020    32730
       12      228     4020    65040  1047540
       30      990    32730  1047540 33554370

Andrew Howroyd, Table of n, a(n) for n = 1..1275

First and last columns are A027375. Main diagonal is A265627.
Cf. A000031, A000740, A001037, A179043, A323351.
Cf. A323860, A323861, A323863, A323864, A323865, A323867, A323869.

nn=5;
a[n_,k_]:=Sum[MoebiusMu[d]*MoebiusMu[e]*2^(n/d*k/e),{d,Divisors[n]},{e,Divisors[k]}];
Table[a[n-k,k],{n,nn},{k,n-1}]

A(n,k) = {sumdiv(n, d, sumdiv(k,e, moebius(d) * moebius(e) * 2^((n/d) * (k/e))))} \\ Andrew Howroyd, Jan 19 2023

A(n,k) = Sum_{d|n, e|k} mu(d) * mu(e) * 2^((n/d) * (k/e)).

A323872 Number of n X n aperiodic binary toroidal necklaces.

Original entry on oeis.org

1, 2, 2, 54, 4050, 1342170, 1908852102, 11488774559598, 288230375950387200, 29850020237398244599296, 12676506002282260237970435130, 21970710674130840874443091905460038, 154866286100907105149455216472736043777350, 4427744605404865645682169434028029029963535277450

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

The 1-dimensional (Lyndon word) case is A001037.

We define a toroidal necklace to be an equivalence class of matrices under all possible rotations of the sequence of rows and the sequence of columns. An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			Inequivalent representatives of the a(2) = 2 aperiodic necklaces:
  [0 0] [0 1]
  [0 1] [1 1]
Inequivalent representatives of the a(3) = 54 aperiodic necklaces:
  000  000  000  000  000  000  000  000  000
  000  000  001  001  001  001  001  001  001
  001  011  001  010  011  100  101  110  111
.
  000  000  000  000  000  000  000  000  000
  011  011  011  011  011  011  011  111  111
  001  010  011  100  101  110  111  001  011
.
  001  001  001  001  001  001  001  001  001
  001  001  001  001  001  001  010  010  010
  010  011  100  101  110  111  011  101  110
.
  001  001  001  001  001  001  001  001  001
  010  011  011  011  011  011  100  100  100
  111  010  011  101  110  111  011  110  111
.
  001  001  001  001  001  001  001  001  001
  101  101  101  101  110  110  110  110  111
  011  101  110  111  011  101  110  111  011
.
  001  001  001  011  011  011  011  011  011
  111  111  111  011  011  011  101  110  111
  101  110  111  101  110  111  111  111  111

Andrew Howroyd, Table of n, a(n) for n = 0..50

Main diagonal of A323861.
Cf. A000031, A000740, A001037, A027375, A059966, A179043, A184271, A323351.
Cf. A323859, A323860, A323865, A323866, A323871.

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
Table[Length[Select[(Partition[#,n]&)/@Tuples[{0,1},n^2],And[apermatQ[#],neckmatQ[#]]&]],{n,4}]

Terms a(5) and beyond from Andrew Howroyd, Aug 21 2019

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

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

cp's OEIS Frontend

A376822 Number of colorings of a toroidal n X n grid using exactly two colors under translational symmetry.

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A376823 Number of colorings of a toroidal n X n grid using exactly three colors under translational symmetry.

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A376824 Number of colorings of a toroidal n X n grid using exactly four colors under translational symmetry.

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A323862 Table read by antidiagonals where A(n,k) is the number of n X k binary arrays in which both the sequence of rows and the sequence of columns are (independently) aperiodic.

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A323872 Number of n X n aperiodic binary toroidal necklaces.

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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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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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