cp's OEIS Frontend

A004075 Number of Skolem sequences of order n.

%I A004075 #45 Oct 24 2023 06:22:21
%S A004075 1,0,0,6,10,0,0,504,2656,0,0,455936,3040560,0,0,1400156768,
%T A004075 12248982496,0,0,11435578798976,123564928167168,0,0,
%U A004075 204776117691241344,2634563519776965376,0,0,7064747252076429464064,105435171495207196553472,0,0
%N A004075 Number of Skolem sequences of order n.
%C A004075 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.
%C A004075 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
%D A004075 CRC Handbook of Combinatorial Designs, 1996, p. 460.
%H A004075 Ali Assarpour, Amotz Bar-Noy, Ou Liuo, <a href="http://arxiv.org/abs/1507.00315">Counting the Number of Langford Skolem Pairings</a>, arXiv:1507.00315 [cs.DM], 2015.
%H A004075 S. Burrill and L. Yen, <a href="http://arxiv.org/abs/1301.6424">Constructing Skolem sequences via generating trees</a>, arXiv preprint arXiv:1301.6424 [math.CO], 2013.
%H A004075 J. E. Miller, <a href="http://dialectrix.com/langford.html">Langford's Problem</a>
%H A004075 G. Nordh, <a href="http://arxiv.org/abs/math/0506155">Perfect Skolem sequences</a>, arXiv:math/0506155 [math.CO], 2005.
%F A004075 For n > 1, a(n) = A059106(n)*2 because A059106 ignores reflected solutions. - _Martin Fuller_, Mar 08 2007
%t A004075 (* Program not suitable to compute a large number of terms. *)
%t A004075 iter[n_] := Sequence @@ Table[{x[i], {-1, 1}}, {i, 1, 2n}];
%t A004075 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];
%t A004075 Table[Print[a[n]]; a[n], {n, 1, 10}] (* _Jean-François Alcover_, Sep 29 2018, from formula in Assarpour et al. *)
%Y A004075 Cf. A014552, A059106, A176127, A268537.
%K A004075 nonn,more
%O A004075 1,4
%A A004075 _N. J. A. Sloane_
%E A004075 More terms (via A059106) from _Martin Fuller_, Mar 08 2007
%E A004075 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_
%E A004075 a(28)-a(31) from Assarpour et al. (2015), added by _Max Alekseyev_, Sep 24 2023