A184331 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..6 arrays.

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

1,1

Links

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.

Formula

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

Example

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

Mathematica

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 *)

Prog

(PARI)
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

Crossrefs

Columns 1-3 are A054626, A184329, A184330.

Cf. A184271, A184284.

Sequence in context: A300529 A299468 A200974 * A267432 A155712 A015817

Adjacent sequences: A184328 A184329 A184330 * A184332 A184333 A184334

Keyword

nonn,tabl

Author

R. H. Hardin, Jan 11 2011

Status

approved