A184277 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..3 arrays.

4, 10, 10, 24, 76, 24, 70, 700, 700, 70, 208, 8296, 29184, 8296, 208, 700, 104968, 1398500, 1398500, 104968, 700, 2344, 1399176, 71582944, 268447936, 71582944, 1399176, 2344, 8230, 19175140, 3817765120, 54975633976, 54975633976

Offset

1,1

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 48 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 pp. 3, 42.

Formula

T(n,k) = (1/(n*k)) * Sum_{c|n} Sum_{d|k} phi(c) * phi(d) * 4^(n*k/lcm(c,d)). - Andrew Howroyd, Sep 27 2017

Example

Table starts
      4        10           24             70            208            700
     10        76          700           8296         104968        1399176
     24       700        29184        1398500       71582944     3817765120
     70      8296      1398500      268447936    54975633976 11728126132976
    208    104968     71582944    54975633976 45035996274688
    700   1399176   3817765120 11728126132976
   2344  19175140 209430787824
   8230 268447816
  29144

Mathematica

T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*4^(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) * 4^(n*k/lcm(c, d)))); \\ Andrew Howroyd, Sep 27 2017

Crossrefs

Columns 1-5 are A001868, A184273, A184274, A184275, A184276.

Main diagonal is A184272.

Cf. A184271, A184284.

Sequence in context: A310337 A201836 A201216 * A264163 A223165 A263558

Adjacent sequences: A184274 A184275 A184276 * A184278 A184279 A184280

Keyword

nonn,tabl

Author

R. H. Hardin, Jan 10 2011

Status

approved