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%I #97 Feb 25 2024 09:09:58
%S 0,0,1,1,0,0,26,150,0,0,17792,108144,0,0,39809640,326721800,0,0,
%T 256814891280,2636337861200,0,0,3799455942515488,46845158056515936,0,
%U 0,111683611098764903232,1607383260609382393152,0,0
%N Number of solutions to Langford (or Langford-Skolem) problem (up to reversal of the order).
%C These are also called Langford pairings.
%C 2*a(n) = A176127(n) gives the number of ways of arranging the numbers 1,1,2,2,...,n,n so that there is one number between the two 1's, two numbers between the two 2's, ..., n numbers between the two n's.
%C a(n) > 0 iff n == 0 or 3 (mod 4).
%D Jaromir Abrham, "Exponential lower bounds for the numbers of Skolem and extremal Langford sequences," Ars Combinatoria 22 (1986), 187-198.
%D M. Gardner, Mathematical Magic Show, New York: Vintage, pp. 70 and 77-78, 1978.
%D M. Gardner, Mathematical Magic Show, Revised edition published by Math. Assoc. Amer. in 1989. Contains a postscript on pp. 283-284 devoted to a discussion of early computations of the number of Langford sequences.
%D R. K. Guy, The unity of combinatorics, Proc. 25th Iranian Math. Conf, Tehran, (1994), Math. Appl 329 129-159, Kluwer Dordrecht 1995, Math. Rev. 96k:05001.
%D D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 2.
%D M. Krajecki, Christophe Jaillet and Alain Bui, "Parallel tree search for combinatorial problems: A comparative study between OpenMP and MPI," Studia Informatica Universalis 4 (2005), 151-190.
%D Roselle, David P. Distributions of integers into s-tuples with given differences. Proceedings of the Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1971), pp. 31--42. Dept. Comput. Sci., Univ. Manitoba, Winnipeg, Man., 1971. MR0335429 (49 #211). - From _N. J. A. Sloane_, Jun 05 2012
%H Ali Assarpour, Amotz Bar-Noy, and 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 Gheorghe Coserea, <a href="/A014552/a014552.txt">Solutions for n=7</a>.
%H Gheorghe Coserea, <a href="/A014552/a014552_1.txt">Solutions for n=8</a>.
%H Gheorghe Coserea, <a href="/A014552/a014552_1.mzn.txt">MiniZinc model for generating solutions</a>.
%H R. O. Davies, <a href="http://www.jstor.org/stable/3610650">On Langford's problem II</a>, Math. Gaz., 1959, vol. 43, 253-255.
%H Elin Farnell, <a href="https://doi.org/10.1080/10511970.2016.1195465">Puzzle Pedagogy: A Use of Riddles in Mathematics Education</a>, PRIMUS, July 2016, pp. 202-211.
%H M. Krajecki, <a href="http://dialectrix.com/langford/krajecki/krajecki-letter.html">L(2,23)=3,799,455,942,515,488</a>.
%H C. D. Langford, <a href="http://www.jstor.org/stable/3610392">2781. Parallelograms with Integral Sides and Diagonals</a>, Math. Gaz., 1958, vol. 42, p. 228.
%H J. E. Miller, <a href="http://dialectrix.com/langford.html">Langford's Problem</a>
%H G. Nordh, <a href="http://arxiv.org/abs/math/0506155">Perfect Skolem sequences</a>, arXiv:math/0506155 [math.CO], 2005.
%H Zan Pan, <a href="https://eprint.panzan.me/articles/langford.pdf">Conjectures on the number of Langford sequences</a>, (2021).
%H Michael Penn, <a href="https://www.youtube.com/watch?v=jzjgMBdpvow">Why is this list "nice"? -- Langford's Problem</a>, YouTube video, 2022.
%H C. J. Priday, <a href="http://www.jstor.org/stable/3610650">On Langford's Problem I</a>, Math. Gaz., 1959, vol. 43, 250-255.
%H W. Schneider, <a href="http://web.archive.org/web/2004/www.wschnei.de/digit-related-numbers/langfords-problem.html">Langford's Problem</a>
%H T. Skolem, <a href="http://www.mscand.dk/article/view/10490">On certain distributions of integers in pairs with given differences</a>, Math. Scand., 1957, vol. 5, 57-68.
%H T. Saito and S. Hayasaka, <a href="http://www.jstor.org/stable/3618042">Langford sequences: a progress report</a>, Math. Gaz., 1979, vol. 63, #426, 261-262.
%H J. E. Simpson, <a href="http://dx.doi.org/10.1016/0012-365X(83)90008-0">Langford Sequences: perfect and hooked</a>, Discrete Math., 1983, vol. 44, #1, 97-104.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LangfordsProblem.html">Langford's Problem</a>.
%F a(n) = A176127(n)/2.
%e Solutions for n=3 and 4: 312132 and 41312432.
%e Solution for n=16: 16, 14, 12, 10, 13, 5, 6, 4, 15, 11, 9, 5, 4, 6, 10, 12, 14, 16, 13, 8, 9, 11, 7, 1, 15, 1, 2, 3, 8, 2, 7, 3.
%Y See A050998 for further examples of solutions.
%Y If the zeros are omitted we get A192289.
%Y Cf. A059106, A059107, A059108, A125762, A026272.
%K nonn,hard,nice,more
%O 1,7
%A John E. Miller (john@timehaven.us), _Eric W. Weisstein_, _N. J. A. Sloane_
%E a(20) from Ron van Bruchem and Mike Godfrey, Feb 18 2002
%E a(21)-a(23) sent by John E. Miller (john@timehaven.us) and Pab Ter (pabrlos(AT)yahoo.com), May 26 2004. These values were found by a team at Université de Reims Champagne-Ardenne, headed by Michael Krajecki, using over 50 processors for 4 days.
%E a(24)=46845158056515936 was computed circa Apr 15 2005 by the Krajecki team. - _Don Knuth_, Feb 03 2007
%E Edited by _Max Alekseyev_, May 31 2011
%E a(27) from the J. E. Miller web page "Langford's problem"; thanks to _Eric Desbiaux_ for reporting this. - _N. J. A. Sloane_, May 18 2015. However, it appears that the value was wrong. - _N. J. A. Sloane_, Feb 22 2016
%E Corrected and 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_.
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