A184284 -id:A184284 - 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

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

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

A184278 Number of distinct n X n toroidal 0..2 arrays.

Original entry on oeis.org

3, 27, 2211, 2691711, 33891544611, 4169295457320267, 4883659780216684279491, 53651309692070594433108320631, 5474401089420219382077156686715199875, 5153775207320113310364632834595843483435609099, 44553974378043749018508590814287728257805180848046070883

Offset: 1

R. H. Hardin, Jan 10 2011

nonn

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

Diagonal of A184284.

a(n) = (1/(n^2)) * Sum_{c|n} Sum_{d|n} phi(c) * phi(d) * 3^(n^2/lcm(c,d)). - Andrew Howroyd, Sep 27 2017

Terms a(6) and beyond from Andrew Howroyd, Sep 27 2017

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

Original entry on oeis.org

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

A184279 Number of distinct n X 2 toroidal 0..2 arrays.

Original entry on oeis.org

6, 27, 130, 855, 5934, 44487, 341802, 2691675, 21524542, 174348099, 1426419858, 11767964475, 97764131646, 817028814447, 6863038218842, 57906884938095, 490505347906086, 4169295457231347, 35548729443032994, 303941636737938639

Offset: 1

R. H. Hardin, Jan 10 2011

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200

Column 2 of A184284.

Terms a(17) and beyond from Andrew Howroyd, Sep 27 2017

A184280 Number of distinct n X 3 toroidal 0..2 arrays.

Original entry on oeis.org

11, 130, 2211, 44368, 956635, 21524790, 498112275, 11767920118, 282429542331, 6863038218842, 168456380815459, 4169295435797688, 103911670590311931, 2605214027190054462, 65651393478909968211, 1661800897440729720268, 42229293393638392639051

Offset: 1

R. H. Hardin, Jan 10 2011

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100

Column 3 of A184284.

Terms a(12) and beyond from Andrew Howroyd, Sep 27 2017

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.

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

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

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A184278 Number of distinct n X n toroidal 0..2 arrays.

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

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Mathematica

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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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