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.
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
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.
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
KEYWORD
nonn,hard
AUTHOR
Ed Pegg Jr, Sep 12 2019
EXTENSIONS
E<=1 proved by Ed Pegg Jr, Oct 16 2019
STATUS
approved