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A138371
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Count of post-period decimal digits up to which the rounded n-th convergent to A058265 agrees with the exact value.
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1
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0, 1, 2, 5, 8, 7, 10, 11, 10, 12, 15, 17, 17, 17, 20, 21, 22, 23, 25, 26, 28, 30, 29, 30, 31, 31, 32, 32, 34, 35, 35, 36, 36, 38, 40, 40, 42, 42, 42, 43, 44, 43, 45, 46, 47, 47, 49, 52, 51, 52, 54, 54, 55, 57, 59, 59, 60, 60, 60, 61, 61, 62, 62, 64, 64, 66, 67, 69, 71, 73, 74
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OFFSET
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1,3
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COMMENTS
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This is a measure of the quality of the n-th convergent to the tribonacci constant A058265 if the convergent and the exact value are compared rounded to an increasing number of digits. The sequence of rounded values of A058265 is 2, 1.8, 1.84, 1.839, 1.8393, 1.83929, 1.839287, 1.8392868, etc. The n-th convergents are 2 (n=1), 11/6 (n=2), 46/25 (n=3), 103/56 (n=4), 31451/17105 (n=5) etc., each with associated rounded decimal expansions.
a(n) is the maximum number of post-period digits of the two expansions if compared at the same level of rounding. Counting only post-period digits (which is one less than the full number of decimal digits) is just a convention taken from A084407.
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LINKS
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EXAMPLE
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For n=4, the 4th convergent is 103/56 = 1.83928571..., with a sequence of rounded representations 2, 1.8, 1.84, 1.839, 1.8393, 1.83929, 1.839286, 1.8392857 etc.
Rounded to 1, 2, 3, 4 or 5 post-period decimal digits, this is the same as the rounded version of the exact value, but disagrees if both are rounded to 6 decimal digits, where 1.839287 <> 1.839286.
So a(4) = 5 (digits), the maximum rounding level with agreement.
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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EXTENSIONS
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Definition and values replaced as defined via continued fractions by R. J. Mathar, Oct 01 2009
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STATUS
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approved
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