A184277 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..3 arrays.
4, 10, 10, 24, 76, 24, 70, 700, 700, 70, 208, 8296, 29184, 8296, 208, 700, 104968, 1398500, 1398500, 104968, 700, 2344, 1399176, 71582944, 268447936, 71582944, 1399176, 2344, 8230, 19175140, 3817765120, 54975633976, 54975633976
Offset: 1
Examples
Table starts
4 10 24 70 208 700
10 76 700 8296 104968 1399176
24 700 29184 1398500 71582944 3817765120
70 8296 1398500 268447936 54975633976 11728126132976
208 104968 71582944 54975633976 45035996274688
700 1399176 3817765120 11728126132976
2344 19175140 209430787824
8230 268447816
29144
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 48 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.
- Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023. See pp. 3, 42.
Crossrefs
Programs
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Mathematica
T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*4^(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) * 4^(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) * 4^(n*k/lcm(c,d)). - Andrew Howroyd, Sep 27 2017