A184284 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..2 arrays.
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
Examples
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
Links
- 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.
Crossrefs
Programs
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Mathematica
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 *) -
PARI
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
Formula
T(n,k) = (1/(n*k)) * Sum_{c|n} Sum_{d|k} phi(c) * phi(d) * 3^(n*k/lcm(c,d)). - Andrew Howroyd, Sep 27 2017