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

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

1,1

Links

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.

Formula

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

Example

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

Maple

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

Mathematica

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 *)

Prog

(PARI)
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

Crossrefs

Columns 1-3 are A054627, A184292, A184293.

Cf. A184271, A184284.

Sequence in context: A089698 A133887 A200713 * A057345 A048740 A139608

Adjacent sequences: A184291 A184292 A184293 * A184295 A184296 A184297

Keyword

nonn,tabl

Author

R. H. Hardin, Jan 10 2011

Status

approved