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A326499
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a(n) = A046693(n) - A309407(n). Excess E of a length n sparse ruler.
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6
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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
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OFFSET
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50
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COMMENTS
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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.
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.
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LINKS
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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nonn,hard
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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