%I A014552
%S A014552 0,0,1,1,0,0,26,150,0,0,17792,108144,0,0,39809640,326721800,0,0,
%T A014552 256814891280,2636337861200,0,0,3799455942515488,46845158056515936
%N A014552 Number of solutions to Langford (or Langford-Skolem) problem.
%C A014552 How many ways are of arranging the numbers 1,1,2,2,3,3,...,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?
%D A014552 Jaromir Abrham, "Exponential lower bounds for the numbers of Skolem and 
               extremal Langford sequences," Ars Combinatoria 22 (1986), 187-198.
%D A014552 R. O. Davies, On Langford's problem II, Math. Gaz., 1959, vol. 43, 253-255.
%D A014552 M. Gardner, Mathematical Magic Show, New York: Vintage, pp. 70 and 77-78, 
               1978.
%D A014552 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 A014552 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 A014552 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 A014552 C. D. Langford, Math. Gaz., 1958, vol. 42, p. 228.
%D A014552 C. J. Priday, On Langford's Problem I, Math. Gaz., 1959, vol. 43, 250-253.
%D A014552 Saito and Hayasaka, Langford sequences: a progress report, Math. Gaz., 
               1979, vol. 63, #426, 261-262.
%D A014552 J. E. Simpson, Langford Sequences: perfect and hooked, Discrete Math., 
               1983, vol. 44, #1, 97-104.
%D A014552 T. Skolem, On certain distributions of integers in pairs with given differences, 
               Math. Scand., 1957, vol. 5, 57-68.
%H A014552 Michael Krajecki, <a href="http://www.lclark.edu/~miller/langford/L2,
               23.html">L(2,23)=3,799,455,942,515,488</a>.
%H A014552 J. E. Miller, <a href="http://www.lclark.edu/~miller/langford.html">Langford's 
               Problem</a>
%H A014552 J. E. Miller, <a href="http://www.lclark.edu/~miller/langford/L2,23.html">
               Latest report on Langford's problem</a>
%H A014552 G. Nordh, <a href="http://arXiv.org/abs/math.NT/0506155">Perfect Skolem 
               sequences</a>
%H A014552 W. Schneider, <a href="http://www.wschnei.de/digit-related-numbers/langfords-problem.html">
               Langford's Problem</a>
%H A014552 T. Skolem, <a href="http://134.76.163.65/servlet/digbib?template=view.html&id=179852&startpage=60&endpage=71&\
               image-path=http://134.76.176.141/cgi-bin/letgifsfly.cgi&image-subpath=/
               4698&image-subpath=4698&pagenumber=60&imageset-id=4698">On certain 
               distributions of integers in pairs with given differences</a>, Math. 
               Scand., 1957, vol. 5, 57-68.
%H A014552 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LangfordsProblem.html">Langford's Problem</a>.
%F A014552 a(n) > 0 iff n == -1 or 0 mod 4.
%e A014552 Solutions for n=3 and 4: 312132 and 41312432. 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 A014552 See A050998 for further examples of solutions. Cf. A059106, A059107, 
               A059108, A125762.
%Y A014552 Sequence in context: A042316 A042318 A166801 this_sequence A042314 A027280 
               A006354
%Y A014552 Adjacent sequences: A014549 A014550 A014551 this_sequence A014553 A014554 
               A014555
%K A014552 nonn,hard,nice
%O A014552 1,7
%A A014552 John Miller (miller(AT)lclark.edu), Eric Weisstein (eric(AT)weisstein.com), 
               N. J. A. Sloane (njas(AT)research.att.com).
%E A014552 a(20) from Ron van Bruchem and Mike Godfrey, Feb 18, 2002
%E A014552 a(21)-a(23) sent by John Miller (miller(AT)lclark.edu) and Pab Ter (pabrlos(AT)yahoo.com), 
               May 26 2004. These values were found by a team at Universite de Reims 
               Champagne-Ardenne, headed by Michael Krajecki, using over 50 processors 
               for 4 days.
%E A014552 a(24)=46845158056515936 was computed circa Apr 15 2005 by the Krajecki 
               team. - D. E. Knuth, Feb 03 2007