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%I #41 Oct 11 2020 05:54:26
%S 1,1,3,3,5,7,13,15,27,25,45,59,89,103,163,187,281,313,469,533,835,873,
%T 1319,1551,2093,2347,3477,3881,5363,5871,8267,9443,12887,14069,19229,
%U 22113,29359,32229,44127,48659,64789,71167,94625,105699,139119,151145,199657
%N Number of Golomb rulers of length n.
%C Wanted: a recurrence. Are any of A169940-A169954 related to any other entries in the OEIS?
%C 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
%C 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
%H Fausto A. C. Cariboni, <a href="/A169942/b169942.txt">Table of n, a(n) for n = 1..99</a>
%H T. Pham, <a href="http://math.sfsu.edu/beck/teach/masters/tu.pdf">Enumeration of Golomb Rulers</a> (Master's thesis), San Francisco State U., 2011.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GolombRuler.html">Golomb Ruler.</a>
%H <a href="/index/Ca#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>
%H <a href="/index/Go#Golomb">Index entries for sequences related to Golomb rulers</a>
%F a(n) = A169952(n) - A169952(n-1) for n>1. - _Andrew Howroyd_, Jul 09 2017
%e 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
%e From _Gus Wiseman_, May 16 2019: (Start)
%e The a(1) = 1 through a(8) = 15 compositions such that every restriction to a subinterval has a different sum:
%e (1) (2) (3) (4) (5) (6) (7) (8)
%e (12) (13) (14) (15) (16) (17)
%e (21) (31) (23) (24) (25) (26)
%e (32) (42) (34) (35)
%e (41) (51) (43) (53)
%e (132) (52) (62)
%e (231) (61) (71)
%e (124) (125)
%e (142) (143)
%e (214) (152)
%e (241) (215)
%e (412) (251)
%e (421) (341)
%e (512)
%e (521)
%e (End)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@ReplaceList[#,{___,s__,___}:>Plus[s]]&]],{n,15}] (* _Gus Wiseman_, May 16 2019 *)
%o (Sage)
%o def A169942(n):
%o R = QQ['x']
%o return sum(1 for c in cartesian_product([[0, 1]]*n) if max(R([1] + list(c) + [1])^2) == 2)
%o [A169942(n) for n in range(1,8)]
%o # _Tomas Boothby_, May 15 2012
%Y Related to thickness: A169940-A169954, A061909.
%Y Related to Golomb rulers: A036501, A054578, A143823.
%Y Row sums of A325677.
%Y Cf. A000079, A103295, A103300, A108917, A143824, A325466, A325545, A325676, A325678, A325679, A325683, A325686.
%K nonn
%O 1,3
%A _N. J. A. Sloane_, Aug 01 2010
%E a(15)-a(30) from _Nathaniel Johnston_, Nov 12 2011
%E a(31)-a(50) from _Tomas Boothby_, May 15 2012
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