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A014552
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Number of solutions to Langford (or Langford-Skolem) problem (up to reversal of the order).
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10
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0, 0, 1, 1, 0, 0, 26, 150, 0, 0, 17792, 108144, 0, 0, 39809640, 326721800, 0, 0, 256814891280, 2636337861200, 0, 0, 3799455942515488, 46845158056515936, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,7
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COMMENTS
| These are also called Langford pairings.
2*a(n)=A176127(n) gives the number of 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.
a(n) > 0 iff n == 0 or 3 mod 4.
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REFERENCES
| Jaromir Abrham, "Exponential lower bounds for the numbers of Skolem and extremal Langford sequences," Ars Combinatoria 22 (1986), 187-198.
R. O. Davies, On Langford's problem II, Math. Gaz., 1959, vol. 43, 253-255.
M. Gardner, Mathematical Magic Show, New York: Vintage, pp. 70 and 77-78, 1978.
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.
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. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 2.
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.
C. D. Langford, Math. Gaz., 1958, vol. 42, p. 228.
C. J. Priday, On Langford's Problem I, Math. Gaz., 1959, vol. 43, 250-253.
Saito and Hayasaka, Langford sequences: a progress report, Math. Gaz., 1979, vol. 63, #426, 261-262.
J. E. Simpson, Langford Sequences: perfect and hooked, Discrete Math., 1983, vol. 44, #1, 97-104.
T. Skolem, On certain distributions of integers in pairs with given differences, Math. Scand., 1957, vol. 5, 57-68.
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LINKS
| Michael Krajecki, L(2,23)=3,799,455,942,515,488.
J. E. Miller, Langford's Problem
J. E. Miller, Latest report on Langford's problem
G. Nordh, Perfect Skolem sequences
W. Schneider, Langford's Problem
T. Skolem, On certain distributions of integers in pairs with given differences, Math. Scand., 1957, vol. 5, 57-68.
Eric Weisstein's World of Mathematics, Langford's Problem.
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FORMULA
| a(n) = A176127(n)/2
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EXAMPLE
| 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.
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CROSSREFS
| See A050998 for further examples of solutions.
If the zeros are omitted we get A192289.
Cf. A059106, A059107, A059108, A125762, A026272.
Sequence in context: A042318 A166801 A200449 * A192289 A042314 A200874
Adjacent sequences: A014549 A014550 A014551 * A014553 A014554 A014555
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KEYWORD
| nonn,hard,nice,more
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AUTHOR
| John Miller (miller(AT)lclark.edu), Eric Weisstein (eric(AT)weisstein.com), N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| a(20) from Ron van Bruchem and Mike Godfrey, Feb 18, 2002
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.
a(24)=46845158056515936 was computed circa Apr 15 2005 by the Krajecki team. - D. E. Knuth, Feb 03 2007
Edited by Max Alekseyev (maxale(AT)gmail.com), May 31 2011
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