A169942 Number of Golomb rulers of length n.

1, 1, 3, 3, 5, 7, 13, 15, 27, 25, 45, 59, 89, 103, 163, 187, 281, 313, 469, 533, 835, 873, 1319, 1551, 2093, 2347, 3477, 3881, 5363, 5871, 8267, 9443, 12887, 14069, 19229, 22113, 29359, 32229, 44127, 48659, 64789, 71167, 94625, 105699, 139119, 151145, 199657

Offset: 1

Author

N. J. A. Sloane, Aug 01 2010

Keywords

• nonn

Comments

Wanted: a recurrence. Are any of A169940-A169954 related to any other entries in the OEIS?

Leading entry in row n of triangle in A169940. Also the number of Sidon sets A with min(A) = 0 and max(A) = n. Odd for all n since {0,n} is the only symmetric Golomb ruler, and reversal preserves the Golomb property. Bounded from above by A032020 since the ruler {0 < r_1 < ... < r_t < n} gives rise to a composition of n: (r_1 - 0, r_2 - r_1, ... , n - r_t) with distinct parts. - Tomas Boothby, May 15 2012

Also the number of compositions of n such that every restriction to a subinterval has a different sum. This is a stronger condition than all distinct consecutive subsequences having a different sum (cf. A325676). - Gus Wiseman, May 16 2019

Examples

			For n=2, there is one Golomb Ruler: {0,2}.  For n=3, there are three: {0,3}, {0,1,3}, {0,2,3}. - _Tomas Boothby_, May 15 2012
From _Gus Wiseman_, May 16 2019: (Start)
The a(1) = 1 through a(8) = 15 compositions such that every restriction to a subinterval has a different sum:
  (1)  (2)  (3)   (4)   (5)   (6)    (7)    (8)
            (12)  (13)  (14)  (15)   (16)   (17)
            (21)  (31)  (23)  (24)   (25)   (26)
                        (32)  (42)   (34)   (35)
                        (41)  (51)   (43)   (53)
                              (132)  (52)   (62)
                              (231)  (61)   (71)
                                     (124)  (125)
                                     (142)  (143)
                                     (214)  (152)
                                     (241)  (215)
                                     (412)  (251)
                                     (421)  (341)
                                            (512)
                                            (521)
(End)
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 1..99
• T. Pham, Enumeration of Golomb Rulers (Master's thesis), San Francisco State U., 2011.
• Eric Weisstein's World of Mathematics, Golomb Ruler.
• Index entries for sequences related to carryless arithmetic
• Index entries for sequences related to Golomb rulers

Crossrefs

Related to thickness: A169940-A169954, A061909.

Related to Golomb rulers: A036501, A054578, A143823.

Row sums of A325677.

Cf. A000079, A103295, A103300, A108917, A143824, A325466, A325545, A325676, A325678, A325679, A325683, A325686.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@ReplaceList[#,{_,s__,_}:>Plus[s]]&]],{n,15}] (* Gus Wiseman, May 16 2019 *)

Sage

def A169942(n):
    R = QQ['x']
    return sum(1 for c in cartesian_product([[0, 1]]*n) if max(R([1] + list(c) + [1])^2) == 2)
[A169942(n) for n in range(1,8)]
# Tomas Boothby, May 15 2012

Formula

a(n) = A169952(n) - A169952(n-1) for n>1. - Andrew Howroyd, Jul 09 2017

Extensions

a(15)-a(30) from Nathaniel Johnston, Nov 12 2011

a(31)-a(50) from Tomas Boothby, May 15 2012