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A004137
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Maximal number of edges in a graceful graph on n nodes.
(Formerly M2526)
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18
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0, 1, 3, 6, 9, 13, 17, 23, 29, 36, 43, 50, 58, 68, 79, 90, 101, 112, 123, 138, 153, 168, 183, 198, 213, 232
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OFFSET
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1,3
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COMMENTS
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A graph with e edges is "graceful" if its nodes can be labeled with distinct integers in {0,1,...,e} so that, if each edge is labeled with the absolute difference between the labels of its endpoints, then the e edges have the distinct labels 1, 2, ..., e.
Equivalently, maximum m for which there's a restricted difference basis with respect to m with n elements. A "difference basis w.r.t. m" is a set of integers such that every integer from 1 to m is a difference between two elements of the set. A "restricted" difference basis is one in which the smallest element is 0 and the largest is m.
a(n) is also the length of an optimal ruler with n marks. For definitions see A103294. For example, a(6)=13 is the length of the optimal rulers with 6 marks, {[0, 1, 6, 9, 11, 13], [0, 2, 4, 7, 12, 13], [0, 1, 4, 5, 11, 13], [0, 2, 8, 9, 12, 13], [0, 1, 2, 6, 10, 13], [0, 3, 7, 11, 12, 13]}. Also n = 1 + A103298(a(n)). - Peter Luschny, Feb 28 2005
If the conjecture is true that an optimal ruler with more than 12 segments is a Wichmann ruler then the sequence continues 232, 251, 270, 289, 308, 327, ... - Peter Luschny, Oct 09 2011 [updated to take the verifications of Robison into account, Oct 01 2015]
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REFERENCES
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J.-C. Bermond, Graceful graphs, radio antennae and French windmills, pp. 18-37 of R. J. Wilson, editor, Graph Theory and Combinatorics. Pitman, London, 1978.
R. K. Guy, Modular difference sets and error correcting codes. in: Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, chapter C10, (2004), 181-183.
J. C. P. Miller, Difference bases: Three problems in additive number theory, pp. 299-322 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Al Zimmermann's Programming Contests, Graceful Graphs, September - December 2013.
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FORMULA
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EXAMPLE
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a(7)=17: Label the 7 nodes 0,1,8,11,13,15,17 and include all edges except those from 8 to 15, from 13 to 15, from 13 to 17 and from 15 to 17. {0,1,8,11,13,15,17} is a restricted difference basis w.r.t. 17.
a(21)=153 because there exists a complete ruler (i.e., one that can measure every distance between 1 and 153) with marks [0,1,2,3,7,14,21,28,43,58,73,88,103,118,126,134,142,150,151,152,153] and no complete ruler of greater length with the same number of marks can be found. This ruler is of the type described by B. Wichmann and it is conjectured by Peter Luschny that it is impossible to improve upon Wichmann's construction for finding optimal rulers of bigger lengths.
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PROG
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(C) See Klaus Nagel link.
(Parallel C++) See A. Robison link.
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CROSSREFS
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A080060 is an erroneous version of the sequence, given in Bermond's paper. Cf. A005488.
A289761 provides the conjectured continuation.
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KEYWORD
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nonn,nice,hard,more
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AUTHOR
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EXTENSIONS
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Miller's paper gives these lower bounds for the 8 terms from a(15) to a(22): 79, 90, 101, 112, 123, 138, 153, 168.
Using this program Hugo Pfoertner found further evidence for the conjectured term a(21)=153, Feb 23 2005
Terms a(21) .. a(24) proved by exhaustive search by Arch D. Robison, Hugo Pfoertner, Nov 01 2013
Term a(25) proved by exhaustive search by Arch D. Robison, Peter Luschny, Jan 14 2014
Term a(26) proved by exhaustive search by Fabian Schwartau, Yannic Schröder, Lars Wolf, Joerg Schoebel, Feb 22 2021
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STATUS
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approved
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