A234943 -id:A234943 - cp's OEIS frontend

A003022 Length of shortest (or optimal) Golomb ruler with n marks.

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, 585

Offset: 2

a(n) is the least integer such that there is an n-element 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 n-mark Golomb ruler has a unique integer distance between any pair of marks and thus measures n(n-1)/2 distinct integer distances.
An optimal n-mark Golomb ruler has the smallest possible length (distance between the two end marks) for an n-mark ruler.
A perfect n-mark Golomb ruler has length exactly n(n-1)/2 and measures each distance from 1 to n(n-1)/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 length-n 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)

			a(5)=11 because 0-1-4-9-11 (0-2-7-10-11) resp. 0-3-4-9-11 (0-2-7-8-11) are shortest: there is no b0-b1-b2-b3-b4 with different distances |bi-bj| and max. |bi-bj| < 11.

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. 23-37 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.
Richard K. Guy, Unsolved Problems in Number Theory (2nd edition), Springer-Verlag (1994), Section C10.
A. Kotzig and P. J. Laufer, Sum triangles of natural numbers having minimum top, Ars. Combin. 21 (1986), 5-13.
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. 299--322. Academic Press, London,1971. MR0316269 (47 #4817)
Rhys Price Jones, Gracelessness, Proc. 10th S.-E. Conf. Combin., Graph Theory and Computing, 1979, pp. 547-552.
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. 136-147.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Anonymous, In Search Of The Optimal 20, 21 and 22 Mark Golomb Rulers
Thomas Bloom, Problem 30, Problem 43, Problem 155, and Problem 861, Erdős Problems.
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.
A. Kotzig and P. J. Laufer, Sum triangles of natural numbers having minimum top, Ars. Combin. 21 (1986), 5-13. [Annotated scanned copy]
Joseph Malkevitch, Weird Rulers.
Greg Martin and Kevin O'Bryant, Constructions of generalized Sidon sets, arXiv:math/0408081 [math.NT], 2004-2005.
Lloyd Miller, Golomb Rulers
Kevin 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
Bill Rankin, Golomb Ruler Calculations
Walter Schneider, Golomb Rulers
James B. Shearer, Golomb ruler table
James B. Shearer, Table of Known Optimal Golomb Rulers
James B. Shearer, Difference Triangle Sets: Known optimal solutions.
James B. Shearer, Difference Triangle Sets: Discoverers
David Singmaster, David Fielker, and N. J. A. Sloane, Correspondence, August 1979
N. J. A. Sloane, First few optimal Golomb rulers
Terence Tao, Erdős problem database, see nos. 30, 43, 155, 861.
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

See A106683 for triangle of marks.
Cf. A008404, A036501, A039953, A078106, A030873.
0-1-4-9-11 corresponds to 1-3-5-2 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.

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

from itertools import combinations, combinations_with_replacement, count
def a(n):
    for k in count(n-1):
        for c in combinations(range(k), n-1):
            c = c + (k, )
            ss = set()
            for s in combinations_with_replacement(c, 2):
                if sum(s) in ss: break
                else: ss.add(sum(s))
            if len(ss) == n*(n+1)//2: return k # Jianing Song, Feb 14 2025, adapted from the python program of A345731

a(n) >= n(n-1)/2, with strict inequality for n >= 5 (Golomb). - David W. Wilson, Aug 18 2007

425 sent by Ed Pegg Jr, Nov 15 2004
a(25), a(26) proved by OGR-25 and OGR-26 projects, added by Max Alekseyev, Sep 29 2010
a(27) proved by OGR-27, added by David Consiglio, Jr., Jun 09 2014
a(28) proved by OGR-28, added by David Consiglio, Jr., Jan 19 2023

A013574 Minimal scope of an (n,2) difference triangle.

Original entry on oeis.org

3, 7, 10, 12, 15, 19, 22, 24, 27, 31, 34, 36, 39, 43, 46, 48, 51, 55, 58, 60, 63, 67, 70, 72, 75, 79, 82, 84, 87, 91, 94, 96, 99, 103, 106, 108, 111, 115, 118, 120, 123, 127, 130, 132, 135, 139, 142, 144, 147, 151, 154, 156, 159, 163, 166, 168, 171, 175, 178, 180, 183, 187, 190

Offset: 1

N. J. A. Sloane

a(n)*Pi is also the total length of irregular spiral (center points: 1, 2, 5, 3, 4) after n-rotations. - Kival Ngaokrajang, Jan 08 2014

CRC Handbook of Combinatorial Designs, 1996, p. 315.

Colin Barker, Table of n, a(n) for n = 1..1000
Y. M. Chee, C. J. Colbourn, Constructions for difference triangle sets, arXiv:0712.2553 [cs.IT], 2007.
Kival Ngaokrajang, Illustration of irregular spiral (center points: 1, 2, 5, 3, 4)
J. B. Shearer, Difference Triangle Sets: Known optimal solutions.
J. B. Shearer, Difference Triangle Sets: Discoverers
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

A row or column of array in A234943.

A319279 is an essentially identical sequence.

A013574 := proc(n)
    if modp(n,4) in {0,1} then
        3*n ;
    else
        3*n+1 ;
    end if;
end proc: # R. J. Mathar, Nov 28 2016

LinearRecurrence[{2, -2, 2, -1}, {3, 7, 10, 12}, 63] (* Jean-François Alcover, Nov 24 2017 *)

Vec(x*(3 + x + 2*x^2) / ((1 - x)^2*(1 + x^2)) + O(x^40)) \\ Colin Barker, Nov 25 2017

a(n) = 3n if n = {0,1} (mod 4). a(n) = 3n+1 if n = {2,3} (mod 4). [Chee Theor. 2] - R. J. Mathar, Nov 28 2016
G.f.: x*(3+x+2*x^2) / ( (x^2+1)*(x-1)^2 ). - R. J. Mathar, Nov 28 2016
From Colin Barker, Nov 25 2017: (Start)
a(n) = (-1/4 - i/4) * ((-1+i) + (-i)^n - i*i^n - (6-6*i)*n).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
(End)

A234944 Array read by antidiagonals: T(i,j) = number of different optimal difference triangle sets M(i,j).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 12, 2, 2, 1, 8, 9, 1, 4, 1, 21, 12, 4, 2, 5, 1, 720, 174, 6, 3, 2, 1, 1, 4079, 3132, 92, 1, 4, 1, 1, 1, 3040, 71930, 117

Offset: 1

N. J. A. Sloane, Jan 08 2014

This array gives the number of different ways to choose the optimal difference triangle sets M(i,j) described in A234943.

			Array begins:
j\i| 1,  2,  3,   4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15,
-----------------------------------------------------------------
1  | 1,   1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,   1,   1 (A000012)
2  | 1,   3,   12,   8,  21, 720, 4079, 3040, 20505, 1669221, 14708525, 10567748,  (A234947)
3  | 1,   2,    9,  12, 174, 3132, 71930,   (A234945)
4  | 2,   1,    4,   6,  92, 117,   (A234946)
5  | 4,   2,    3,   1,
6  | 5,   2,    4,   4,
7  | 1,   1,    1,
8  | 1,   2,
9  | 1,   1,
10 | 2,
11 | 1
12 | 1
13 | 1
14 | 1
15 | 1
The entries match those in A234943.