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A326499
a(n) = A046693(n) - A309407(n). Excess E of a length n sparse ruler.
6
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
J. Leech, On the representation of 1, 2, ..., n by differences, J. Lond. Math. Soc. 31 (1956), 160-169.
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