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

Author

R. H. Hardin, Jan 10 2011

Keywords

• nonn
• tabl

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

Columns 1-6 are A001867, A184279, A184280, A184281, A184282, A184283.

Main diagonal is A184278.

Cf. A184271, A184277, A184288, A184291, A184331, A184294 (0..1, 0..3 etc.).

Programs

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