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A184284 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..2 arrays. 13
3, 6, 6, 11, 27, 11, 24, 130, 130, 24, 51, 855, 2211, 855, 51, 130, 5934, 44368, 44368, 5934, 130, 315, 44487, 956635, 2691711, 956635, 44487, 315, 834, 341802, 21524790, 174342216, 174342216, 21524790, 341802, 834, 2195, 2691675, 498112275 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Alois P. Heinz, antidiagonals n = 1..50, flattened (first 58 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.
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023. See p. 3.
FORMULA
T(n,k) = (1/(n*k)) * Sum_{c|n} Sum_{d|k} phi(c) * phi(d) * 3^(n*k/lcm(c,d)). - Andrew Howroyd, Sep 27 2017
EXAMPLE
Table starts
3 6 11 24 51 130
6 27 130 855 5934 44487
11 130 2211 44368 956635 21524790
24 855 44368 2691711 174342216 11767964475
51 5934 956635 174342216 33891544611 6863038218842
130 44487 21524790 11767964475 6863038218842
315 341802 498112275 817028472960
834 2691675 11767920118
2195 21524542
5934
MATHEMATICA
T[n_, k_] := (1/(n*k))*Sum[EulerPhi[c]*EulerPhi[d]*3^(n*k/LCM[c, d]), {c, Divisors[n]}, {d, Divisors[k]}]; Table[T[n-k+1, k], {n, 1, 9}, {k, 1, n}] // Flatten (*Jean-François Alcover, Oct 07 2017, after Andrew Howroyd *)
PROG
(PARI)
T(n, k) = (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 3^(n*k/lcm(c, d)))); \\ Andrew Howroyd, Sep 27 2017
CROSSREFS
Main diagonal is A184278.
Cf. A184271, A184277, A184288, A184291, A184331, A184294 (0..1, 0..3 etc.).
Sequence in context: A201155 A200534 A138743 * A287882 A066297 A160713
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Jan 10 2011
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)