A184291 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..5 arrays.
6, 21, 21, 76, 351, 76, 336, 7826, 7826, 336, 1560, 210456, 1119936, 210456, 1560, 7826, 6047412, 181402676, 181402676, 6047412, 7826, 39996, 181410426, 31345666736, 176319685116, 31345666736, 181410426, 39996, 210126, 5597460306
Offset: 1
Examples
Table starts
6 21 76 336 1560 7826 39996
21 351 7826 210456 6047412 181410426 5597460306
76 7826 1119936 181402676 31345666736 5642220395616
336 210456 181402676 176319685116
1560 6047412 31345666736
7826 181410426
39996
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]*6^(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) * 6^(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) * 6^(n*k/lcm(c,d)). - Andrew Howroyd, Sep 27 2017