A184284
Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..2 arrays.
Original entry on oeis.org
3, 6, 6, 11, 27, 11, 24, 130, 130, 24, 51, 855, 2211, 855, 51, 130, 5934, 44368, 44368, 5934, 130, 315, 44487, 956635, 2691711, 956635, 44487, 315, 834, 341802, 21524790, 174342216, 174342216, 21524790, 341802, 834, 2195, 2691675, 498112275
Offset: 1
Table starts
3 6 11 24 51 130
6 27 130 855 5934 44487
11 130 2211 44368 956635 21524790
24 855 44368 2691711 174342216 11767964475
51 5934 956635 174342216 33891544611 6863038218842
130 44487 21524790 11767964475 6863038218842
315 341802 498112275 817028472960
834 2691675 11767920118
2195 21524542
5934
- Alois P. Heinz, antidiagonals n = 1..50, flattened (first 58 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 p. 3.
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T[n_, k_] := (1/(n*k))*Sum[EulerPhi[c]*EulerPhi[d]*3^(n*k/LCM[c, d]), {c, Divisors[n]}, {d, Divisors[k]}]; Table[T[n-k+1, k], {n, 1, 9}, {k, 1, n}] // Flatten (*Jean-François Alcover, Oct 07 2017, after Andrew Howroyd *)
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T(n, k) = (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 3^(n*k/lcm(c,d)))); \\ Andrew Howroyd, Sep 27 2017
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
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.
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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
A184272
Number of distinct n X n toroidal 0..3 arrays.
Original entry on oeis.org
4, 76, 29184, 268447936, 45035996274688, 131176846752109387776, 6467605103205252048452583424, 5316911983139663492479919370381869056, 72172920362019897195243695442781187577923764224, 16069380442589902755419620923449655543228876820136712994816
Offset: 1
-
T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*4^(n*k/LCM[c, d]), {d, Divisors[k]}], {c, Divisors[n]}]; a[n_] := T[n, n]; Array[a, 10] (* Jean-François Alcover, Oct 31 2017, after Andrew Howroyd *)
A184273
Number of distinct n X 2 toroidal 0..3 arrays.
Original entry on oeis.org
10, 76, 700, 8296, 104968, 1399176, 19175140, 268447816, 3817763740, 54975738736, 799645010860, 11728126132976, 173215372864600, 2573485530115576, 38430716856090160, 576460752706084936, 8680820741074491040, 131176846752106589576, 1988364834899715824380
Offset: 1
-
T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*4^(n*k/LCM[c, d]), {d, Divisors[k]}], {c, Divisors[n]}]; a[n_] := T[n, 2]; Array[a, 20] (* Jean-François Alcover, Oct 31 2017, after Andrew Howroyd *)
A184274
Number of distinct n X 3 toroidal 0..3 arrays.
Original entry on oeis.org
24, 700, 29184, 1398500, 71582944, 3817765120, 209430787824, 11728124734500, 667199944873344, 38430716856090160, 2235968978631715024, 131176846748291651840, 7749524484709164817824, 460543169377106318541400, 27509778650786228522002432
Offset: 1
-
T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*4^(n*k/LCM[c, d]), {d, Divisors[k]}], {c, Divisors[n]}]; a[n_] := T[n, 3]; Array[a, 15] (* Jean-François Alcover, Oct 31 2017, after Andrew Howroyd *)
A184275
Number of distinct n X 4 toroidal 0..3 arrays.
Original entry on oeis.org
70, 8296, 1398500, 268447936, 54975633976, 11728126132976, 2573485510942780, 576460752706101376, 131176846748288854980, 30223145490448192851232, 7033750223212787748164020, 1650586719047191292234702976, 390046338531762979375904093800
Offset: 1
-
T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*4^(n*k/LCM[c, d]), {d, Divisors[k]}], {c, Divisors[n]}]; a[n_] := T[n, 4]; Array[a, 13] (* Jean-François Alcover, Oct 31 2017, after Andrew Howroyd *)
A184276
Number of distinct n X 5 toroidal 0..3 arrays.
Original entry on oeis.org
208, 104968, 71582944, 54975633976, 45035996274688, 38430716856090160, 33731189163354610720, 30223145490393217217464, 27509778650786228378836960, 25353012004564610547932704768, 23601349356976489220593165438048, 22153799929748597900945512768786000
Offset: 1
-
T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*4^(n*k/LCM[c, d]), {d, Divisors[k]}], {c, Divisors[n]}]; a[n_] := T[n, 5]; Array[a, 12] (* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)
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a(n) = my(k=5); (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 4^(n*k/lcm(c, d)))); \\ Michel Marcus, Nov 01 2017
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