

A003022


Length of shortest (or optimal) Golomb ruler with n marks.
(Formerly M2540)


48



1, 3, 6, 11, 17, 25, 34, 44, 55, 72, 85, 106, 127, 151, 177, 199, 216, 246, 283, 333, 356, 372, 425, 480, 492, 553
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OFFSET

2,2


COMMENTS

a(n) is the least integer such that there is an nelement set of integers between 0 and a(n), the sums of pairs (of not necessarily distinct elements) of which are distinct.
From David W. Wilson, Aug 17 2007: (Start)
An nmark Golomb ruler has a unique integer distance between any pair of marks and thus measures n(n1)/2 distinct integer distances.
An optimal nmark Golomb ruler has the smallest possible length (distance between the two end marks) for an nmark ruler.
A perfect nmark Golomb ruler has length exactly n(n1)/2 and measures each distance from 1 to n(n1)/2. (End)
Positions where A143824 increases (see also A227590).  N. J. A. Sloane, Apr 08 2016
From Gus Wiseman, May 17 2019: (Start)
Also the smallest m such that there exists a lengthn composition of m for which every restriction to a subinterval has a different sum. Representatives of compositions for the first few terms are:
0: ()
1: (1)
3: (2,1)
6: (2,3,1)
11: (3,1,5,2)
17: (4,2,3,7,1)
Representatives of corresponding Golomb rulers are:
{0}
{0,1}
{0,2,3}
{0,2,5,6}
{0,3,4,9,11}
{0,4,6,9,16,17}
(End)


REFERENCES

CRC Handbook of Combinatorial Designs, 1996, p. 315.
A. K. Dewdney, Computer Recreations, Scientific Amer. 253 (No. 6, Jun), 1985, pp. 16ff; 254 (No. 3, March), 1986, pp. 20ff.
S. W. Golomb, How to number a graph, pp. 2337 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.
Richard K. Guy, Unsolved Problems in Number Theory (2nd edition), SpringerVerlag (1994), Section C10.
A. Kotzig and P. J. Laufer, Sum triangles of natural numbers having minimum top, Ars. Combin. 21 (1986), 513.
Miller, J. C. P., Difference bases. Three problems in additive number theory. Computers in number theory (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969), pp. 299322. Academic Press, London,1971. MR0316269 (47 #4817)
Rhys Price Jones, Gracelessness, Proc. 10th S.E. Conf. Combin., Graph Theory and Computing, 1979, pp. 547552.
Ana Salagean, David Gardner and Raphael Phan, Index Tables of Finite Fields and Modular Golomb Rulers, in Sequences and Their Applications  SETA 2012, Lecture Notes in Computer Science. Volume 7280, 2012, pp. 136147.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Table of n, a(n) for n=2..27.
Anonymous, In Search Of The Optimal 20, 21 and 22 Mark Golomb Rulers
A. K. Dewdney, Computer Recreations, Scientific Amer. 253 (No. 6, Jun), 1985, pp. 16ff; 254 (No. 3, March), 1986, pp. 20ff. [Annotated scanned copy]
Distributed.Net, Project OGR
Kent Freeman, Unpublished notes. [Scanned copy]
Michael Geißer, Theresa Körner, Sascha Kurz, and Anne Zahn, Squares with three digits, arXiv:2112.00444 [math.NT], 2021.
S. W. Golomb, Letter to N. J. A. Sloane, 1972.
Google Scholar, Golomb Ruler
A. Kotzig and P. J. Laufer, Sum triangles of natural numbers having minimum top, Ars. Combin. 21 (1986), 513. [Annotated scanned copy]
Joseph Malkevitch, Weird Rulers.
G. Martin and K. O'Bryant, Constructions of generalized Sidon sets, arXiv:math/0408081 [math.NT], 20042005.
L. Miller, Golomb Rulers
K. O'Bryant, Sets of Natural Numbers with Proscribed Subsets, J. Int. Seq. 18 (2015) # 15.7.7
Ed Pegg, Jr., Math Games: Rulers, Arrays, and Gracefulness
B. Rankin, Golomb Ruler Calculations
W. Schneider, Golomb Rulers
J. B. Shearer, Golomb ruler table
J. B. Shearer, Table of Known Optimal Golomb Rulers
J. B. Shearer, Difference Triangle Sets: Known optimal solutions.
J. B. Shearer, Difference Triangle Sets: Discoverers
David Singmaster, David Fielker, N. J. A. Sloane, Correspondence, August 1979
N. J. A. Sloane, First few optimal Golomb rulers
D. Vanderschel et al., In Search Of The Optimal 20, 21 and 22 Mark Golomb Rulers
Eric Weisstein's World of Mathematics, Golomb Ruler.
Wikipedia, Golomb ruler
Index entries for sequences related to Golomb rulers


FORMULA

a(n) >= n(n1)/2, with strict inequality for n >= 5 (Golomb).  David W. Wilson, Aug 18 2007


EXAMPLE

a(5)=11 because 014911 (0271011) resp. 034911 (027811) are shortest: there is no b0b1b2b3b4 with different distances bibj and max. bibj < 11.


MATHEMATICA

Min@@Total/@#&/@GatherBy[Select[Join@@Permutations/@Join@@Table[IntegerPartitions[i], {i, 0, 15}], UnsameQ@@ReplaceList[#, {___, s__, ___}:>Plus[s]]&], Length] (* Gus Wiseman, May 17 2019 *)


CROSSREFS

See A106683 for triangle of marks.
Cf. A008404, A036501, A039953, A078106, A030873.
014911 corresponds to 1352 in A039953: 0+1+3+5+2=11
A row or column of array in A234943.
Adding 1 to these terms gives A227590. Cf. A143824.
For first differences see A270813.
Cf. A103295, A108917, A143823, A169942.
Cf. A325466, A325545, A325676, A325677, A325678, A325683.
Sequence in context: A173143 A109413 A294397 * A025722 A022775 A025743
Adjacent sequences: A003019 A003020 A003021 * A003023 A003024 A003025


KEYWORD

nonn,hard,nice,more


AUTHOR

N. J. A. Sloane


EXTENSIONS

425 sent by Ed Pegg Jr, Nov 15 2004
a(25), a(26) proved by OGR25 and OGR26 projects, added by Max Alekseyev, Sep 29 2010
a(27) proved by OGR27, added by David Consiglio, Jr., Jun 09 2014


STATUS

approved



