A184288 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..4 arrays.

5, 15, 15, 45, 175, 45, 165, 2635, 2635, 165, 629, 49075, 217125, 49075, 629, 2635, 976887, 20346485, 20346485, 976887, 2635, 11165, 20349075, 2034505661, 9536816875, 2034505661, 20349075, 11165, 48915, 435970995, 211927741375

Offset

1,1

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 39 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) * 5^(n*k/lcm(c,d)). - Andrew Howroyd, Sep 27 2017

Example

Table starts
      5        15           45           165           629         2635
     15       175         2635         49075        976887     20349075
     45      2635       217125      20346485    2034505661 211927741375
    165     49075     20346485    9536816875 4768372070757
    629    976887   2034505661 4768372070757
   2635  20349075 211927741375
  11165 435970995
  48915

Mathematica

T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c] * EulerPhi[d] * 5^(n*k/LCM[c, d]), {d, Divisors[k]}], {c, Divisors[n]}];
Table[T[n - k + 1, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 31 2017, after Andrew Howroyd *)

Prog

(PARI)
T(n, k) = (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 5^(n*k/lcm(c, d)))); \\ Andrew Howroyd, Sep 27 2017

Crossrefs

Columns 1-4 are A001869, A184286, A184287, A184288.

Cf. A184271, A184284.

Sequence in context: A330082 A160275 A200858 * A015666 A291627 A045856

Adjacent sequences: A184285 A184286 A184287 * A184289 A184290 A184291

Keyword

nonn,tabl

Author

R. H. Hardin, Jan 10 2011

Status

approved