A036501 Number of inequivalent Golomb rulers with n marks and shortest length.

1, 1, 1, 2, 4, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• nice
• more

Comments

From Gus Wiseman, May 31 2019: (Start)

A Golomb ruler of length n is a subset of {0..n} containing 0 and n and such that every pair of distinct terms has a different difference. For example, the a(2) = 1 through a(8) = 1 Golomb rulers are:

2: {0,1}

3: {0,1,3}

4: {0,1,4,6}

5: {0,1,4,9,11}

5: {0,2,7,8,11}

6: {0,1,4,10,12,17}

6: {0,1,4,10,15,17}

6: {0,1,8,11,13,17}

6: {0,1,8,12,14,17}

7: {0,1,4,10,18,23,25}

7: {0,1,7,11,20,23,25}

7: {0,2,3,10,16,21,25}

7: {0,2,7,13,21,22,25}

7: {0,1,11,16,19,23,25}

8: {0,1,4,9,15,22,32,34}

Also half the number of length-(n - 1) compositions of A003022(n) such that every consecutive subsequence has a different sum. For example, the a(2) = 1 through a(8) = 1 compositions are (A = 10):

2: (1)

3: (1,2)

4: (1,3,2)

5: (1,3,5,2)

5: (2,5,1,3)

6: (1,3,6,2,5)

6: (1,3,6,5,2)

6: (1,7,3,2,4)

6: (1,7,4,2,3)

7: (1,3,6,8,5,2)

7: (1,6,4,9,3,2)

7: (2,1,7,6,5,4)

7: (2,5,6,8,1,3)

7: (1,A,5,3,4,2)

8: (1,3,5,6,7,A,2)

(End)

Links

• Anonymous, In Search Of The Optimal 20, 21 and 22 Mark Golomb Rulers
• Distributed.Net, Project OGR
• J. B. Shearer, Table of known optimal Golomb rulers
• Eric Weisstein's World of Mathematics, Golomb Ruler
• Index entries for sequences related to Golomb rulers

Crossrefs

Cf. A003022, A039953, A054578, A108917, A143823, A169942, A270813, A325677, A325683.