A326499 a(n) = A046693(n) - A309407(n). Excess E of a length n sparse ruler.

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1

Offset

50

Comments

Excess = minimal length n sparse ruler marks - round(sqrt(3*n + 9/4)).

The first fifty terms are 0. A308766 lists n values with excess 1.

Luschny's conjecture: 1^3 24^1 5^1 4^5 3^2 with length 58 is the last non-Wichmann optimal ruler. If this is true, all terms are 0 or 1.

Taking terms in batches based on A289761 leads to pattern illustrated at A046693.

Terms over n = 213 are unverified minimal.

"Dark Satanic Mills on a Cloudy Day." - N. J. A. Sloane

This is a hard sequence due to minimality verification. For example, n=474 has E=1, but it's possible an E=0 sparse ruler exists.

Links

Table of n, a(n) for n=50..150.

J. Leech, On the representation of 1, 2, ..., n by differences, J. Lond. Math. Soc. 31 (1956), 160-169.

Peter Luschny, Perfect and Optimal Rulers

Peter Luschny, Are optimal rulers of Wichmann type?

Peter Luschny, Perfect Rulers.

Peter Luschny, Wichmann Rulers.

Ed Pegg Jr., Sparse Rulers (Wolfram Demonstrations Project)

Ed Pegg Jr., Wichmann-like Rulers (Wolfram Demonstrations Project)

Ed Pegg Jr, Table of n, a(n) for n=1..10501 in batches of A289761. Transpose for Dark Mills pattern.

Ed Pegg Jr, Picture of a(n) for n = 1..10501 in batches of A289761. This is the Dark Mills pattern.

L. Rédei, A. Rényi, On the representation of the numbers 1, 2, ..., N by means of differences, Matematicheskii Sbornik, Vol. 24(66) Num. 3 (1949), 385-389 (in Russian).

Arch D. Robison, Parallel Computation of Sparse Rulers, Jan 14 2014.

B. Wichmann, A note on restricted difference bases, J. Lond. Math. Soc. 38 (1963), 465-466.

Ed Pegg Jr, Sparse rulers and excess values for lengths n=1..10501.

Example

0, 1, 2, 3, 4, 10, 16, 22, 28, 34, 40, 46, 51 is a sparse ruler of length 51 with 13 marks, the fewest possible. 13 - round(sqrt(3*51+9/4)) = 13 - 12 = 1.

Crossrefs

Cf. A046693, A289761, A308766, A309407.

Sequence in context: A014057 A015689 A359160 * A104124 A347246 A052434

Adjacent sequences: A326496 A326497 A326498 * A326500 A326501 A326502

Keyword

nonn,hard

Author

Ed Pegg Jr, Sep 12 2019

Extensions

E<=1 proved by Ed Pegg Jr, Oct 16 2019

Status

approved