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
Examples
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
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.
Programs
-
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 *) -
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
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