A179043 -id:A179043 - cp's OEIS frontend

A270858 Number of n X n checkered tori (see A179043) with at least one 0 and at least one 1 in each row and each column.

0, 1, 14, 1446, 705366, 1304451482

Offset: 1

Krasimir Yordzhev, Mar 24 2016

Let us denote by Qn the set of all n X n binary matrices with at least one 0 and at least one 1 in each row and each column. Let A,B belong to Qn. We will say that A and B are equivalent if B is obtained from A as a result of a sequential cyclic move of the last row or column at a first position. [Please rewrite that sentence!] It's easy to see that this defines an equivalence relation. The sequence gives the number of equivalence classes. The elements of the factor-set Qn/~ are relevant in the textile industry.

K. Yordzhev, An example for the use of bitwise operations in programming, Mathematics and education in mathematics, v. 38 (2009), 196-202, also arXiv:1201.1468 [cs.OH], 2012.
K. Yordzhev, H. Kostadinova, Mathematical Modelling in the Textile Industry Bulletin of Mathematical Sciences & Applications, Vol. 1, No. 1 (2012), pp. 20 -35.

Cf. A179043.

A184271 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal binary arrays (n >= 1, k >= 1).

Original entry on oeis.org

2, 3, 3, 4, 7, 4, 6, 14, 14, 6, 8, 40, 64, 40, 8, 14, 108, 352, 352, 108, 14, 20, 362, 2192, 4156, 2192, 362, 20, 36, 1182, 14624, 52488, 52488, 14624, 1182, 36, 60, 4150, 99880, 699600, 1342208, 699600, 99880, 4150, 60, 108, 14602, 699252, 9587580, 35792568

Offset: 1

R. H. Hardin, Jan 10 2011

This is a 2-dimensional generalization of binary necklaces (A000031). A toroidal array or necklace can be defined either as an equivalence class of matrices under all possible rotations of the sequence of rows and the sequence of columns, or as a matrix that is minimal among all possible rotations of its sequence of rows and its sequence of columns. - Gus Wiseman, Feb 04 2019

			      1     2        3           4            5             6              7
----------------------------------------------------------------------------
1:    2     3        4           6            8            14             20
2:    3     7       14          40          108           362           1182
3:    4    14       64         352         2192         14624          99880
4:    6    40      352        4156        52488        699600        9587580
5:    8   108     2192       52488      1342208      35792568      981706832
6:   14   362    14624      699600     35792568    1908897152   104715443852
7:   20  1182    99880     9587580    981706832  104715443852 11488774559744
8:   36  4150   699252   134223976  27487816992 5864063066500
9:   60 14602  4971184  1908881900 781874936816
10: 108 52588 35792568 27487869472
From _Gus Wiseman_, Feb 04 2019: (Start)
Inequivalent representatives of the T(2,3) = 14 toroidal necklaces:
  [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 1] [0 0 1] [0 0 1]
  [0 0 0] [0 0 1] [0 1 1] [1 1 1] [0 0 1] [0 1 0] [0 1 1]
.
  [0 0 1] [0 0 1] [0 0 1] [0 1 1] [0 1 1] [0 1 1] [1 1 1]
  [1 0 1] [1 1 0] [1 1 1] [0 1 1] [1 0 1] [1 1 1] [1 1 1]
(End)

Alois P. Heinz, Antidiagonals n = 1..100, flattened (first 95 terms from R. H. Hardin)
S. N. Ethier, Counting toroidal binary arrays, arXiv preprint arXiv:1301.2352, 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.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023.
Wikipedia, Pólya enumeration theorem

Columns 1-8 are A000031, A184264, A184265, A184266, A184267, A184268, A184269, A184270.
Main diagonal is A179043.
Cf. A001037 (binary Lyndon words), A008965, A323858, A323859 (binary toroidal necklaces of size n), A323861 (aperiodic version), A323865, A323870 (normal toroidal necklaces), A323872.

a[n_, k_] := Sum[If[Mod[n, c] == 0, Sum[If[Mod[k, d] == 0, EulerPhi[c] EulerPhi[d] 2^(n k/LCM[c, d]), 0], {d, 1, k}], 0], {c, 1, n}]/(n k)
(* second program *)
neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
Table[Length[Select[Partition[#,n-k]&/@Tuples[{0,1},(n-k)*k],neckmatQ]],{n,8},{k,n-1}] (* Gus Wiseman, Feb 04 2019 *)

T(n,k) = (1/(nk))*Sum_{ c divides n } Sum_{ d divides k } phi(c)*phi(d)*2^(nk/lcm(c,d)), where phi is A000010 and lcm stands for least common multiple. - Stewart N. Ethier, Aug 24 2012

A061417 Number of permutations up to cyclic rotations; permutation siteswap necklaces.

Original entry on oeis.org

1, 2, 4, 10, 28, 136, 726, 5100, 40362, 363288, 3628810, 39921044, 479001612, 6227066928, 87178295296, 1307675013928, 20922789888016, 355687438476444, 6402373705728018, 121645100594641896, 2432902008177690360, 51090942175425331320, 1124000727777607680022

Offset: 1

Antti Karttunen, May 02 2001

If permutations are converted to (i,p(i)) permutation arrays, then this automorphism is obtained by their "SW-NE diagonal toroidal shifts" (see Matthias Engelhardt's Java program in A006841), while the Maple procedure below converts each permutation to a siteswap pattern (used in juggling), rotates it by one digit and converts the resulting new (or same) siteswap pattern back to a permutation.
When the subset of permutations listed by A064640 are subjected to the same automorphism one gets A002995.
The number of conjugacy classes of the symmetric group of degree n when conjugating only with the cyclic permutation group of degree n. - Attila Egri-Nagy, Aug 15 2014
Also the number of equivalence classes of permutations of {1...n} under the action of rotation of vertices in the cycle decomposition. The corresponding action on words applies m -> m + 1 for m < n and n -> 1, and rotates once to the right. For example, (24531) first becomes (35142) under the application of cyclic rotation, and then is rotated right to give (23514). - Gus Wiseman, Mar 04 2019

			If I have a five-element permutation like 25431, in cycle notation (1 2 5)(3 4), I mark the numbers 1-5 clockwise onto a circle and draw directed edges from 1 to 2, from 2 to 5, from 5 to 1 and a double-way edge between 3 and 4. All the 5-element permutations that produce some rotation (discarding the labels of the nodes) of that chord diagram belong to the same equivalence class with 25431. The sequence gives the count of such equivalence classes.

Cf. A006841, A060495. For other Maple procedures, see A060501 (Perm2SiteSwap2), A057502 (CountCycles), A057509 (rotateL), A060125 (PermRank3R and permul).
A061417[p] = A061860[p] = (p-1)!+(p-1) for all prime p's.
A064636 (derangements-the same automorphism).
A061417[n] = A064649[n]/n.
Cf. A000031, A000939, A002995, A008965, A060223, A064640, A086675 (digraphical necklaces), A179043, A192332, A275527 (path necklaces), A323858, A323859, A323870, A324513, A324514 (aperiodic permutations).

List([1..10],n->Size( OrbitsDomain( CyclicGroup(IsPermGroup,n), SymmetricGroup( IsPermGroup,n),\^))); # Attila Egri-Nagy, Aug 15 2014

a061417 = sum . a047917_row  -- Reinhard Zumkeller, Mar 19 2014

Algebraic formula: with(numtheory); SSRPCC := proc(n) local d,s; s := 0; for d in divisors(n) do s := s + phi(n/d)*((n/d)^d)*(d!); od; RETURN(s/n); end;
Empirically: with(group); SiteSwapRotationPermutationCycleCounts := proc(upto_n) local b,u,n,a,r; a := []; for n from 1 to upto_n do b := []; u := n!; for r from 0 to u-1 do b := [op(b),1+PermRank3R(SiteSwap2Perm1(rotateL(Perm2SiteSwap2(PermUnrank3Rfix(n,r)))))]; od; a := [op(a),CountCycles(b)]; od; RETURN(a); end;
PermUnrank3Rfixaux := proc(n,r,p) local s; if(0 = n) then RETURN(p); else s := floor(r/((n-1)!)); RETURN(PermUnrank3Rfixaux(n-1, r-(s*((n-1)!)), permul(p,[[n,n-s]]))); fi; end;
PermUnrank3Rfix := (n,r) -> convert(PermUnrank3Rfixaux(n,r,[]),'permlist',n);
SiteSwap2Perm1 := proc(s) local e,n,i,a; n := nops(s); a := []; for i from 1 to n do e := ((i+s[i]) mod n); if(0 = e) then e := n; fi; a := [op(a),e]; od; RETURN(convert(invperm(convert(a,'disjcyc')),'permlist',n)); end;

a[n_] := (1/n)*Sum[ EulerPhi[n/d]*(n/d)^d*d!, {d, Divisors[n]}]; Table[a[n], {n, 1, 21}] (* Jean-François Alcover, Oct 09 2012, from formula *)
Table[Length[Select[Permutations[Range[n]],#==First[Sort[NestList[RotateRight[#/.k_Integer:>If[k==n,1,k+1]]&,#,n-1]]]&]],{n,8}] (* Gus Wiseman, Mar 04 2019 *)

a(n) = (1/n)*sumdiv(n, d, eulerphi(n/d)*(n/d)^d*d!); \\ Indranil Ghosh, Apr 10 2017

from sympy import divisors, factorial, totient
def a(n):
    return sum(totient(n//d)*(n//d)**d*factorial(d) for d in divisors(n))//n
print([a(n) for n in range(1, 22)]) # Indranil Ghosh, Apr 10 2017

a(n) = (1/n)*Sum_{d|n} phi(n/d)*((n/d)^d)*(d!).

A323858 Number of toroidal necklaces of positive integers summing to n.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 31, 44, 90, 154, 296, 524, 1035, 1881, 3636, 6869, 13208, 25150, 48585, 93188, 180192, 347617, 673201, 1303259, 2529740, 4910708, 9549665, 18579828, 36192118, 70540863, 137620889, 268655549, 524873503, 1026068477, 2007178821, 3928564237

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

The 1-dimensional (necklace) case is A008965.

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. Alternatively, a toroidal necklace is a matrix that is minimal among all possible rotations of its sequence of rows and its sequence of columns.

			Inequivalent representatives of the a(6) = 31 toroidal necklaces:
  6  15  24  33  114  123  132  222  1113  1122  1212  11112  111111
.
  1  2  3  11  11  12  12  111
  5  4  3  13  22  12  21  111
.
  1  1  1  2  11
  1  2  3  2  11
  4  3  2  2  11
.
  1  1  1
  1  1  2
  1  2  1
  3  2  2
.
  1
  1
  1
  1
  2
.
  1
  1
  1
  1
  1
  1

Andrew Howroyd, Table of n, a(n) for n = 0..200
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.

Cf. A000031, A000670, A008965, A059966, A101509, A179043, A184271.

Cf. A323859, A323861, A323865, A323866, A323870.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
Table[Length[Join@@Table[Select[ptnmats[k],neckmatQ],{k,Times@@Prime/@#&/@IntegerPartitions[n]}]],{n,10}]

U(n,m,k) = (1/(n*m)) * sumdiv(n, c, sumdiv(m, d, eulerphi(c) * eulerphi(d) * subst(k, x, x^lcm(c,d))^(n*m/lcm(c, d))));
a(n)={if(n < 1, n==0, sum(i=1, n, sum(j=1, n\i, polcoef(U(i, j, x/(1-x) + O(x*x^n)), n))))} \\ Andrew Howroyd, Aug 18 2019

Terms a(18) and beyond from Andrew Howroyd, Aug 18 2019

A323861 Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary toroidal necklaces.

Original entry on oeis.org

2, 1, 1, 2, 2, 2, 3, 9, 9, 3, 6, 27, 54, 27, 6, 9, 99, 335, 335, 99, 9, 18, 326, 2182, 4050, 2182, 326, 18, 30, 1161, 14523, 52377, 52377, 14523, 1161, 30, 56, 4050, 99858, 698535, 1342170, 698535, 99858, 4050, 56, 99, 14532, 698870, 9586395, 35790267, 35790267, 9586395, 698870, 14532, 99

Offset: 1

Gus Wiseman, Feb 04 2019

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.

			Table begins:
        1    2    3    4
    ------------------------
  1: |  2    1    2    3
  2: |  1    2    9   27
  3: |  2    9   54  335
  4: |  3   27  335 4050
Inequivalent representatives of the A(3,2) = 9 aperiodic toroidal necklaces:
  [0 0 0] [0 0 0] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 1 1] [0 1 1]
  [0 0 1] [0 1 1] [0 1 0] [0 1 1] [1 0 1] [1 1 0] [1 1 1] [1 0 1] [1 1 1]

Andrew Howroyd, Table of n, a(n) for n = 1..1275
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.
Andrew Howroyd, GAP Program Code

First and last columns are A001037. Main diagonal is A323872.
Cf. A000031, A000740, A027375, A059966, A179043, A184271.
Cf. A323859, A323860, A323865, A323866, A323871.

# See link for code.
for n in [1..8] do for k in [1..8] do Print(A323861(n,k), ", "); od; Print("\n"); od; # Andrew Howroyd, Aug 21 2019

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-k]&/@Tuples[{0,1},(n-k)*k],And[apermatQ[#],neckmatQ[#]]&]],{n,6},{k,n-1}]

Terms a(37) and beyond from Andrew Howroyd, Aug 21 2019

A323865 Number of aperiodic binary toroidal necklaces of size n.

Original entry on oeis.org

1, 2, 2, 4, 8, 12, 36, 36, 114, 166, 396, 372, 1992, 1260, 4644, 8728, 20310, 15420, 87174, 55188, 314064, 399432, 762228, 729444, 5589620, 4026522, 10323180, 19883920, 57516048, 37025580, 286322136, 138547332, 805277760, 1041203944, 2021145660, 3926827224

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

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(6) = 36 aperiodic necklaces:
  000001  000011  000101  000111  001011  001101  001111  010111  011111
.
  000  000  001  001  001  001  001  011  011
  001  011  010  011  101  110  111  101  111
.
  00  00  00  00  00  01  01  01  01
  00  01  01  01  11  01  01  10  11
  01  01  10  11  01  10  11  11  11
.
  0  0  0  0  0  0  0  0  0
  0  0  0  0  0  0  0  1  1
  0  0  0  0  1  1  1  0  1
  0  0  1  1  0  1  1  1  1
  0  1  0  1  1  0  1  1  1
  1  1  1  1  1  1  1  1  1

Andrew Howroyd, Table of n, a(n) for n = 0..200
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.

Cf. A000031, A001037, A027375, A179043, A184271, A323351.

Cf. A323858, A323859, A323860, A323861, A323864, 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]]}]];
zaz[n_]:=Join@@(Table[Partition[#,d],{d,Divisors[n]}]&/@Tuples[{0,1},n]);
Table[If[n==0,1,Length[Union[First/@matcyc/@Select[zaz[n],And[apermatQ[#],neckmatQ[#]]&]]]],{n,0,10}]

a(n) = Sum_{d|n} A323861(d, n/d) for n > 0. - Andrew Howroyd, Aug 21 2019

Terms a(19) and beyond from Andrew Howroyd, Aug 21 2019

A323860 Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary arrays.

Original entry on oeis.org

2, 2, 2, 6, 8, 6, 12, 54, 54, 12, 30, 216, 486, 216, 30, 54, 990, 4020, 4020, 990, 54, 126, 3912, 32730, 64800, 32730, 3912, 126, 240, 16254, 261414, 1047540, 1047540, 261414, 16254, 240, 504, 64800, 2097018, 16764840, 33554250, 16764840, 2097018, 64800, 504

Offset: 1

Gus Wiseman, Feb 04 2019

The 1-dimensional case is A027375.

An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			Table begins:
       1     2     3     4
    ------------------------
  1: |  2     2     6    12
  2: |  2     8    54   216
  3: |  6    54   486  4020
  4: | 12   216  4020 64800
The A(2,2) = 8 arrays:
  [0 0] [0 0] [0 1] [0 1] [1 0] [1 0] [1 1] [1 1]
  [0 1] [1 0] [0 0] [1 1] [0 0] [1 1] [0 1] [1 0]
Note that the following are not aperiodic even though their row and column sequences are independently aperiodic:
  [1 0] [0 1]
  [0 1] [1 0]

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

First and last columns are A027375. Main diagonal is A323863.
Cf. A000740, A001037, A179043, A265627, A323351.
Cf. A323861, A323862, A323864, A323865, A323867, A323869.

# See A323861 for code.
for n in [1..8] do for k in [1..8] do Print(n*k*A323861(n,k), ", "); od; Print("\n"); od; # Andrew Howroyd, Aug 21 2019

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
Table[Length[Select[Partition[#,n-k]&/@Tuples[{0,1},(n-k)*k],apermatQ]],{n,8},{k,n-1}]

T(n,k) = n*k*A323861(n,k). - Andrew Howroyd, Aug 21 2019

Terms a(29) and beyond from Andrew Howroyd, Aug 21 2019

A323859 Number of binary toroidal necklaces of size n.

Original entry on oeis.org

1, 2, 6, 8, 19, 16, 56, 40, 152, 184, 432, 376, 2132, 1264, 4728, 8768, 20688, 15424, 87656, 55192, 315128, 399520, 762984, 729448, 5595408, 4026576, 10325712, 19884504, 57527804, 37025584, 286340544, 138547336, 805335364, 1041204704, 2021176512, 3926827328

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

The 1-dimensional (necklace) case is A000031.

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. Alternatively, a toroidal necklace is a matrix that is minimal among all possible rotations of its sequence of rows and its sequence of columns.

			Inequivalent representatives of the a(4) = 19 binary toroidal necklaces:
  [0 0 0 0] [0 0 0 1] [0 0 1 1] [0 1 0 1] [0 1 1 1] [1 1 1 1]
.
  [0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [1 1]
  [0 0] [0 1] [1 1] [0 1] [1 0] [1 1] [1 1]
.
  [0] [0] [0] [0] [0] [1]
  [0] [0] [0] [1] [1] [1]
  [0] [0] [1] [0] [1] [1]
  [0] [1] [1] [1] [1] [1]

Andrew Howroyd, Table of n, a(n) for n = 0..1000
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.

Cf. A000031, A008965, A179043, A184271, A323351.

Cf. A323858, A323859, A323865, A323870.

matcyc[m_]:=Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
Table[If[n==0,1,Length[Union[First/@matcyc/@Join@@(Table[Partition[#,d],{d,Divisors[n]}]&/@Tuples[{0,1},n])]]],{n,0,10}]

U(n, m, k) = (1/(n*m)) * sumdiv(n, c, sumdiv(m, d, eulerphi(c) * eulerphi(d) * k^(n*m/lcm(c, d))))
a(n) = if(n<1, n==0, sumdiv(n, d, U(n/d, d, 2))) \\ Andrew Howroyd, Jan 24 2023

a(n) = (1/n) * Sum_{d|n} Sum_{e|d, f|(n/d)} phi(e) * phi(f) * 2^(n/lcm(d,n/d)). [Ethier]

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

Original entry on oeis.org

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

A323863 Number of n X n aperiodic binary arrays.

Original entry on oeis.org

1, 2, 8, 486, 64800, 33554250, 68718675672, 562949953420302, 18446744060824780800, 2417851639229257812542976, 1267650600228226023797043513000, 2658455991569831745807614120560664598, 22300745198530623141521551172073990303938400

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			The a(2) = 8 arrays are:
  [0 0] [0 0] [0 1] [0 1] [1 0] [1 0] [1 1] [1 1]
  [0 1] [1 0] [0 0] [1 1] [0 0] [1 1] [0 1] [1 0]
Note that the following are not aperiodic even though their row and column sequences are (independently) aperiodic:
  [1 0] [0 1]
  [0 1] [1 0]

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

Cf. A000031, A000740, A027375, A179043, A265627, A323351.

Cf. A323860, A323862, A323864, A323865, A323867, A323869.

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

a(n) = 2^(n^2) - (n+1)*2^n + 2*n if n is prime. - Robert Israel, Feb 04 2019

a(n) = n^2 * A323872(n). - Andrew Howroyd, Aug 21 2019

a(5) from Robert Israel, Feb 04 2019
a(6)-a(7) from Giovanni Resta, Feb 05 2019
Terms a(8) and beyond from Andrew Howroyd, Aug 21 2019

cp's OEIS Frontend

A270858 Number of n X n checkered tori (see A179043) with at least one 0 and at least one 1 in each row and each column.

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A184271 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal binary arrays (n >= 1, k >= 1).

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A061417 Number of permutations up to cyclic rotations; permutation siteswap necklaces.

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A323858 Number of toroidal necklaces of positive integers summing to n.

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A323861 Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary toroidal necklaces.

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A323865 Number of aperiodic binary toroidal necklaces of size n.

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A323860 Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary arrays.

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