cp's OEIS Frontend

Upper bounds for a(14) to a(26): 18, 21, 24, 26, 28, 29, 33, 36, 37, 40, 43, 46, 49. These have been obtained from the results of the Al Zimmermann competition. - Dmitry Kamenetsky, Apr 23 2021

Upper bounds for a(15) to a(18): 20, 22, 24, 27. - Fausto A. C. Cariboni, Jul 16 2022

An (n,k) difference triangle set is a set of n blocks of k integers such that the difference sets of the blocks are all disjoint. The "scope" of such a set is defined to be the maximal element, if all blocks are translated such that their least elements are all 0. T(n,k) lists the minimal scope for which an (n,k) difference triangle set exists. - Charlie Neder, Jun 14 2019

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)

For 4 and 5 marks the analogous sets of marks are 0,1,4,6 and 0,1,4,9,11.

Golomb ruler: Finite set with property that no difference between any two numbers is repeated and largest number is minimized.

See A003022 for further information and additional references.

a(n) >= n^2-n+1 by a volume bound. A difference set construction by Singer shows that equality holds when n-1 is a prime power. When n is a prime power, a difference set construction by Bose shows that a(n) <= n^2-1. By computation, equality holds in the latter bound at least for 7, 11, 13 and 16.

From Fausto A. C. Cariboni, Aug 13 2017: (Start)

Lexicographically first basis that yields a(n) for n=15..18:

a(15) = 255 from {0,1,3,7,15,26,31,53,63,98,107,127,140,176,197}

a(16) = 255 from {0,1,3,7,15,26,31,53,63,98,107,127,140,176,197,215}

a(17) = 273 from {0,1,3,7,15,31,63,90,116,127,136,181,194,204,233,238,255}

a(18) = 307 from {0,1,3,21,25,31,68,77,91,170,177,185,196,212,225,257,269,274}

(End)

Such sets are also known as modular Golomb rulers, or circular Golomb rulers. - Andrey Zabolotskiy, Sep 11 2017

Place n points on integer coordinates of a square grid of dimension L x L, so that no 3 points are collinear and the areas of the triangles formed by the binomial(n,3) possible triples are all different. a(n) is the minimum L for which this can be achieved. (Note that there are (L+1)^2 lattice points.)

The fact that all triangle areas are multiples of 1/2 and the maximum area supported by the grid is L^2/2 provides us with a lower bound for a(n).

The problem can be considered a 2-dimensional extension of a Golomb ruler and scales to higher dimensions. In general, consider k points on integer coordinates of a D-dimensional hypercube, forming binomial(k,D+1) D-simplices of which the volumes are all different. For D = 1, we recognize the Golomb ruler; for D = 2, we have the problem defined above.

Just like for Costas arrays (another 2-D extension of Golomb rulers), no 2 displacement vectors can be identical, as the diagonal of a parallelogram cuts the shape in triangles of identical area.

a(11) <= 24, a(12) <= 29. - Hugo Pfoertner, Nov 05 2018

Equivalently, a(n) is the number of Sidon sets of {1..n} of size k.