OFFSET
1,1
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.
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
EXAMPLE
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Jan 10 2011
STATUS
approved