A184271 -id:A184271 - cp's OEIS frontend

A294685 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly three colors under translational symmetry, 1 <= k <= n.

0, 0, 9, 2, 91, 2022, 9, 738, 43315, 2679246, 30, 5613, 950062, 174184755, 33887517990, 91, 43404, 21480921, 11765865678, 6862930841141, 4169289730628814, 258, 338259, 497812638, 816999710223, 1429469771994078, 2605213713043722909, 4883659745750360600262, 729, 2679228, 11765822365, 57906482267826, 303941554100145501

Offset: 1

Marko Riedel, Nov 06 2017

Colors are not being permuted, i.e., Power Group Enumeration does not apply here.

			Triangle begins:
   0;
   0,     9;
   2,    91,     2022;
   9,   738,    43315,     2679246;
  30,  5613,   950062,   174184755,   33887517990;
  91, 43404, 21480921, 11765865678, 6862930841141, 4169289730628814;
  ...

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.

Andrew Howroyd, Table of n, a(n) for n = 1..820 (first 40 rows)
Marko Riedel et al., Burnside lemma and translational symmetries of the torus.

Main diagonal is A376823.

Cf. A184271, A184284, A294684, A294686, A294687, A294791, A294792, A294793, A294794. T(n,1) is A056283.

T(n,m)=6*sumdiv(n, d, sumdiv(m, e, eulerphi(d) * eulerphi(e) * stirling(n*m/lcm(d,e), 3, 2) ))/(n*m) \\ Andrew Howroyd, Oct 05 2024

T(n,k) = (Q!/(n*k))*(Sum_{d|n} Sum_{f|k} phi(d) phi(f) S(gcd(d,f)*(n/d)*(k/f), Q)) with Q=3 and S(n,k) Stirling numbers of the second kind.

T(n,k) = A184284(n,k) - 3*A184271(n,k) + 3. - Andrew Howroyd, Oct 05 2024

A294686 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly four colors under translational symmetry, 1 <= k <= n.

Original entry on oeis.org

0, 0, 6, 0, 260, 20720, 6, 5112, 1223136, 257706024, 48, 81876, 67769552, 54278580036, 44900438149488, 260, 1223396, 3731753700, 11681058472672, 38403264917970196, 131160169581733489616, 1200, 17815020, 207438938000, 2570217454576416, 33725471278376393424, 460532748521625850986660, 6467585568566200114362823920, 5106, 257706012, 11681057249536, 576229125971686224

Offset: 1

Marko Riedel, Nov 06 2017

Colors are not being permuted, i.e., Power Group Enumeration does not apply here.

			Triangle begins:
    0;
    0,       6;
    0,     260,      20720;
    6,    5112,    1223136,      257706024;
   48,   81876,   67769552,    54278580036,    44900438149488;
  260, 1223396, 3731753700, 11681058472672, 38403264917970196, 131160169581733489616;
  ...

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.

Andrew Howroyd, Table of n, a(n) for n = 1..820 (first 40 rows)
Marko Riedel et al., Burnside lemma and translational symmetries of the torus.

Main diagonal is A376824.
Cf. A294684, A294685, A294687, A294791, A294792, A294793, A294794. T(n,1) is A056284.
Cf. A184271, A184284, A184277.

T(n,m)=my(k=4); k!*sumdiv(n, d, sumdiv(m, e, eulerphi(d) * eulerphi(e) * stirling(n*m/lcm(d,e), k, 2) ))/(n*m) \\ Andrew Howroyd, Oct 05 2024

T(n,k) = (Q!/(n*k))*(Sum_{d|n} Sum_{f|k} phi(d) phi(f) S(gcd(d,f)*(n/d)*(k/f), Q)) with Q=4 and S(n,k) Stirling numbers of the second kind.

T(n,k) = A184277(n,k) - 4*A184284(n,k) + 6*A184271(n,k) - 4. - Andrew Howroyd, Oct 05 2024

A086675 Number of n X n (0,1)-matrices modulo cyclic permutations of the rows.

Original entry on oeis.org

1, 2, 10, 176, 16456, 6710912, 11453291200, 80421421917440, 2305843009750581376, 268650182136584290872320, 126765060022823052739661424640, 241677817415439249618874010960064512, 1858395433210885261795036719974526548094976

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 27 2003

Also the number of digraphical necklaces with n vertices. A digraphical necklace is defined to be a directed graph that is minimal among all n rotations of the vertices. Alternatively, it is an equivalence class of directed graphs under rotation of the vertices. These are a kind of partially labeled digraphs. - Gus Wiseman, Mar 04 2019

			From _Gus Wiseman_, Mar 04 2019: (Start)
Inequivalent representatives of the a(2) = 10 digraphical necklace edge-sets:
  {}
  {(1,1)}
  {(1,2)}
  {(1,1),(1,2)}
  {(1,1),(2,1)}
  {(1,1),(2,2)}
  {(1,2),(2,1)}
  {(1,1),(1,2),(2,1)}
  {(1,1),(1,2),(2,2)}
  {(1,1),(1,2),(2,1),(2,2)}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..57
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.

Cf. A000031 (binary necklaces), A000939 (cycle necklaces), A008965, A060690, A061417 (permutation necklaces), A184271, A192332 (graphical necklaces), A275527 (path necklaces), A323858 (toroidal necklaces), A323870.

Cf. A324463, A324464, A324513, A324514.

Table[Fold[ #1+EulerPhi[ #2] 2^(n^2 /#2)&, 0, Divisors[n]]/n, {n, 16}]
(* second program *)
rotdigra[g_,m_]:=Sort[g/.k_Integer:>If[k==m,1,k+1]];
Table[Length[Select[Subsets[Tuples[Range[n],2]],#=={}||#==First[Sort[Table[Nest[rotdigra[#,n]&,#,j],{j,n}]]]&]],{n,0,4}] (* Gus Wiseman, Mar 04 2019 *)

a(n) = (1/n)*Sum_{ d divides n } phi(d)*2^(n^2/d) for n > 0, a(0) = 1.

More terms from Wouter Meeussen, Jul 29 2003

a(0)=1 prepended by Gus Wiseman, Mar 04 2019

A184277 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..3 arrays.

Original entry on oeis.org

4, 10, 10, 24, 76, 24, 70, 700, 700, 70, 208, 8296, 29184, 8296, 208, 700, 104968, 1398500, 1398500, 104968, 700, 2344, 1399176, 71582944, 268447936, 71582944, 1399176, 2344, 8230, 19175140, 3817765120, 54975633976, 54975633976

Offset: 1

R. H. Hardin, Jan 10 2011

			Table starts
      4        10           24             70            208            700
     10        76          700           8296         104968        1399176
     24       700        29184        1398500       71582944     3817765120
     70      8296      1398500      268447936    54975633976 11728126132976
    208    104968     71582944    54975633976 45035996274688
    700   1399176   3817765120 11728126132976
   2344  19175140 209430787824
   8230 268447816
  29144

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 48 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.
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023. See pp. 3, 42.

Columns 1-5 are A001868, A184273, A184274, A184275, A184276.
Main diagonal is A184272.
Cf. A184271, A184284.

T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*4^(n*k/LCM[c, d]), {d, Divisors[k]}], {c, Divisors[n]}];
Table[T[n-k+1, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 31 2017, after Andrew Howroyd *)

T(n, k) = (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 4^(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) * 4^(n*k/lcm(c,d)). - Andrew Howroyd, Sep 27 2017

A324463 Number of graphical necklaces covering n vertices.

Original entry on oeis.org

1, 0, 1, 2, 15, 156, 4665, 269618, 31573327, 7375159140, 3450904512841, 3240500443884718, 6113078165054644451, 23175001880311842459108, 176546824267008236554238517, 2701847513793569606737940203894, 83036203475880811677609125194805687

Offset: 0

Gus Wiseman, Feb 28 2019

nonn

A graphical necklace is a simple graph that is minimal among all n rotations of the vertices. Alternatively, it is an equivalence class of simple graphs under rotation of the vertices. Covering means there are no isolated vertices. These are a kind of partially labeled graphs.

			Inequivalent representatives of the a(2) = 1 through a(4) = 15 graphical necklaces:
  {{12}}  {{12}{13}}      {{12}{34}}
          {{12}{13}{23}}  {{13}{24}}
                          {{12}{13}{14}}
                          {{12}{13}{24}}
                          {{12}{13}{34}}
                          {{12}{14}{23}}
                          {{12}{24}{34}}
                          {{12}{13}{14}{23}}
                          {{12}{13}{14}{24}}
                          {{12}{13}{14}{34}}
                          {{12}{13}{24}{34}}
                          {{12}{14}{23}{34}}
                          {{12}{13}{14}{23}{24}}
                          {{12}{13}{14}{23}{34}}
                          {{12}{13}{14}{23}{24}{34}}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 15 covering graphical necklaces, radial embedding.
Gus Wiseman, The a(5) = 156 covering graphical necklaces.

Cf. A000031, A002494, A006129, A008965, A184271, A192332 (non-covering case), A323858, A323859, A323870, A324461, A324462, A324464.

rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],#=={}||#==First[Sort[Table[Nest[rotgra[#,n]&,#,j],{j,n}]]]]&]],{n,0,5}]

a(n)={if(n<1, n==0, sumdiv(n, d, eulerphi(n/d)*sum(k=0, d, (-1)^(d-k)*binomial(d,k)*2^(k*(k-1)*n/(2*d) + k*(n/d\2))))/n)} \\ Andrew Howroyd, Aug 19 2019

a(n) = (1/n)*Sum{d|n} phi(n/d) * Sum_{k=0..d} (-1)^(d-k)*binomial(d,k)*2^( k*(k-1)*n/(2*d) + k*(floor(n/(2*d))) ). - Andrew Howroyd, Aug 19 2019

Terms a(7) and beyond from Andrew Howroyd, Aug 19 2019

A324464 Number of connected graphical necklaces with n vertices.

Original entry on oeis.org

1, 0, 1, 2, 13, 148, 4530, 266614, 31451264, 7366255436, 3449652145180, 3240150686268514, 6112883022923529310, 23174784819204929919428, 176546343645071836902594288, 2701845395848698682311893154024, 83036184895986451215378727412638816, 5122922885438069578928905234650082410736

Offset: 0

Gus Wiseman, Mar 01 2019

nonn

A graphical necklace is a simple graph that is minimal among all n rotations of the vertices. Alternatively, it is an equivalence class of simple graphs under rotation of the vertices. These are a kind of partially labeled graphs.

			Inequivalent representatives of the a(2) = 1 through a(4) = 13 graphical necklaces:
  {{12}}  {{12}{13}}      {{12}{13}{14}}
          {{12}{13}{23}}  {{12}{13}{24}}
                          {{12}{13}{34}}
                          {{12}{14}{23}}
                          {{12}{24}{34}}
                          {{12}{13}{14}{23}}
                          {{12}{13}{14}{24}}
                          {{12}{13}{14}{34}}
                          {{12}{13}{24}{34}}
                          {{12}{14}{23}{34}}
                          {{12}{13}{14}{23}{24}}
                          {{12}{13}{14}{23}{34}}
                          {{12}{13}{14}{23}{24}{34}}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 13 connected graphical necklaces.
Gus Wiseman, The a(5) = 148 connected graphical necklaces.

Cf. A000031, A000939, A001187, A006125, A006129, A008965, A184271, A192332, A275527, A323858, A323859, A323870, A324461, A324462, A324463.

rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],Length[csm[#]]<=1,#=={}||#==First[Sort[Table[Nest[rotgra[#,n]&,#,j],{j,n}]]]]&]],{n,0,5}]

\\ B(n,d) is graphs on n*d points invariant under 1/d rotation.
B(n,d)={2^(n*(n-1)*d/2 + n*(d\2))}
D(n,d)={my(v=vector(n, i, B(i,d)), u=vector(n)); for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1, k)*v[k]*u[n-k])); sumdiv(n, e, eulerphi(d*e) * moebius(e) * u[n/e] * e^(n/e-1))}
a(n)={if(n<=1, n==0, sumdiv(n, d, D(n/d,d))/n)} \\ Andrew Howroyd, Jan 24 2023

Terms a(7) and beyond from Andrew Howroyd, Jan 24 2023

A184288 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..4 arrays.

Original entry on oeis.org

5, 15, 15, 45, 175, 45, 165, 2635, 2635, 165, 629, 49075, 217125, 49075, 629, 2635, 976887, 20346485, 20346485, 976887, 2635, 11165, 20349075, 2034505661, 9536816875, 2034505661, 20349075, 11165, 48915, 435970995, 211927741375

Offset: 1

R. H. Hardin, Jan 10 2011

			Table starts
      5        15           45           165           629         2635
     15       175         2635         49075        976887     20349075
     45      2635       217125      20346485    2034505661 211927741375
    165     49075     20346485    9536816875 4768372070757
    629    976887   2034505661 4768372070757
   2635  20349075 211927741375
  11165 435970995
  48915

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 39 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 1-4 are A001869, A184286, A184287, A184288.

Cf. A184271, A184284.

T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c] * EulerPhi[d] * 5^(n*k/LCM[c, d]), {d, Divisors[k]}], {c, Divisors[n]}];
Table[T[n - k + 1, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 31 2017, after Andrew Howroyd *)

T(n, k) = (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 5^(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) * 5^(n*k/lcm(c,d)). - Andrew Howroyd, Sep 27 2017

A184294 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..7 arrays.

Original entry on oeis.org

8, 36, 36, 176, 1072, 176, 1044, 43800, 43800, 1044, 6560, 2098720, 14913536, 2098720, 6560, 43800, 107377488, 5726645688, 5726645688, 107377488, 43800, 299600, 5726689312, 2345624810432, 17592189193216, 2345624810432, 5726689312, 299600

Offset: 1

R. H. Hardin, Jan 10 2011

			Table starts
       8         36           176           1044          6560      43800
      36       1072         43800        2098720     107377488 5726689312
     176      43800      14913536     5726645688 2345624810432
    1044    2098720    5726645688 17592189193216
    6560  107377488 2345624810432
   43800 5726689312
  299600

Alois P. Heinz, Antidiagonals n = 1..65, flattened (first 8 antidiagonals 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 A054627, A184292, A184293.

Cf. A184271, A184284.

with(numtheory):
T:= (n, k)-> add(add(phi(c)*phi(d)*8^(n*k/ilcm(c, d)),
             c=divisors(n)), d=divisors(k))/(n*k):
seq(seq(T(n, 1+d-n), n=1..d), d=1..8);  # Alois P. Heinz, Aug 20 2017

T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*8^(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 Alois P. Heinz *)

T(n, k) = (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 8^(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) * 8^(n*k/lcm(c,d)). - Andrew Howroyd, Sep 27 2017

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

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))])

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A294685 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly three colors under translational symmetry, 1 <= k <= n.

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A294686 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly four colors under translational symmetry, 1 <= k <= n.

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A086675 Number of n X n (0,1)-matrices modulo cyclic permutations of the rows.

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A184277 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..3 arrays.

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A324463 Number of graphical necklaces covering n vertices.

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A324464 Number of connected graphical necklaces with n vertices.

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A184288 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..4 arrays.

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