A184291 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..5 arrays.

6, 21, 21, 76, 351, 76, 336, 7826, 7826, 336, 1560, 210456, 1119936, 210456, 1560, 7826, 6047412, 181402676, 181402676, 6047412, 7826, 39996, 181410426, 31345666736, 176319685116, 31345666736, 181410426, 39996, 210126, 5597460306

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) * 6^(n*k/lcm(c,d)). - Andrew Howroyd, Sep 27 2017

Example

Table starts
      6        21          76          336        1560          7826      39996
     21       351        7826       210456     6047412     181410426 5597460306
     76      7826     1119936    181402676 31345666736 5642220395616
    336    210456   181402676 176319685116
   1560   6047412 31345666736
   7826 181410426
  39996

Mathematica

T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*6^(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) * 6^(n*k/lcm(c, d)))); \\ Andrew Howroyd, Sep 27 2017

Crossrefs

Columns 1-3 are A054625, A184289, A184290.

Cf. A184271, A184284.

Sequence in context: A298266 A302202 A200831 * A102993 A276803 A143416

Adjacent sequences: A184288 A184289 A184290 * A184292 A184293 A184294

Keyword

nonn,tabl

Author

R. H. Hardin, Jan 10 2011

Status

approved