A184331 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..6 arrays.
7, 28, 28, 119, 637, 119, 616, 19684, 19684, 616, 3367, 721525, 4484039, 721525, 3367, 19684, 28249228, 1153450872, 1153450872, 28249228, 19684, 117655, 1153470437, 316504102999, 2077059243301, 316504102999, 1153470437, 117655, 720916
Offset: 1
Examples
Table starts
7 28 119 616 3367 19684
28 637 19684 721525 28249228 1153470437
119 19684 4484039 1153450872 316504102999 90467424400444
616 721525 1153450872 2077059243301
3367 28249228 316504102999
19684 1153470437
117655
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.
Programs
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Mathematica
T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*7^(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 *) -
PARI
T(n, k) = (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 7^(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) * 7^(n*k/lcm(c,d)). - Andrew Howroyd, Sep 27 2017