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A004075 Number of Skolem sequences of order n. 8
1, 0, 0, 6, 10, 0, 0, 504, 2656, 0, 0, 455936, 3040560, 0, 0, 1400156768, 12248982496, 0, 0, 11435578798976, 123564928167168, 0, 0, 204776117691241344, 2634563519776965376, 0, 0, 7064747252076429464064, 105435171495207196553472, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
Number of permutations of the multiset {1,1,2,2,...,n,n} such that the distance between the elements i equals i for every i=1,2,...,n.
Number of super perfect rhythmic tilings of [0,2n-1] with pairs. See A285698 and A285527 for the definition and tilings of triples and quadruples. - Tony Reix, Apr 25 2017
REFERENCES
CRC Handbook of Combinatorial Designs, 1996, p. 460.
LINKS
Ali Assarpour, Amotz Bar-Noy, Ou Liuo, Counting the Number of Langford Skolem Pairings, arXiv:1507.00315 [cs.DM], 2015.
S. Burrill and L. Yen, Constructing Skolem sequences via generating trees, arXiv preprint arXiv:1301.6424 [math.CO], 2013.
J. E. Miller, Langford's Problem
G. Nordh, Perfect Skolem sequences, arXiv:math/0506155 [math.CO], 2005.
FORMULA
For n > 1, a(n) = A059106(n)*2 because A059106 ignores reflected solutions. - Martin Fuller, Mar 08 2007
MATHEMATICA
(* Program not suitable to compute a large number of terms. *)
iter[n_] := Sequence @@ Table[{x[i], {-1, 1}}, {i, 1, 2n}];
a[n_] := 1/2^(2n) Sum[Product[x[i], {i, 1, 2n}] Product[Sum[x[k] x[k+i], {k, 1, 2n-i}], {i, 1, n}], iter[n] // Evaluate];
Table[Print[a[n]]; a[n], {n, 1, 10}] (* Jean-François Alcover, Sep 29 2018, from formula in Assarpour et al. *)
CROSSREFS
Sequence in context: A110460 A159190 A236766 * A202951 A316633 A295052
KEYWORD
nonn,more
AUTHOR
EXTENSIONS
More terms (via A059106) from Martin Fuller, Mar 08 2007
Extended using results from the Assarpour et al. (2015) paper by N. J. A. Sloane, Feb 22 2016 at the suggestion of William Rex Marshall
a(28)-a(31) from Assarpour et al. (2015), added by Max Alekseyev, Sep 24 2023
STATUS
approved

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Last modified July 17 15:56 EDT 2024. Contains 374377 sequences. (Running on oeis4.)