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
Examples
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
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.
Programs
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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 *) -
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
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