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Search: id:A138367
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| A138367 |
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Count of post-period decimal digits up to which the rounded n-th convergent to sqrt(5) agrees with the exact value. |
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+0
8
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| 0, 2, 4, 5, 6, 7, 8, 10, 8, 12, 14, 14, 16, 18, 19, 20, 21, 23, 24, 24, 26, 28, 29, 30, 31, 33, 33, 34, 35, 37, 39, 40, 41, 42, 44, 44, 46, 47, 48, 49, 51, 53, 53, 55, 56, 57, 59, 60, 60, 61, 64, 65, 66, 68, 69, 70, 72, 73, 74, 75, 76, 77, 79, 80, 81, 83, 83, 85, 85, 88, 89, 90, 91, 92
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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This is a measure of the quality of the n-th convergent to A002163 if the
convergent and the exact value are compared rounded to an increasing
number of digits. The sequence of rounded values of sqrt(5) is
2, 2.2, 2.24, 2.236, 2.2361, 2.23607, 2.236068, 2.2360680 etc, and the n-th convergent
(provided by A001077 and A001076) is to be represented by its equivalent sequence.
a(n) represents the maximum number of post-period digits of the two sequences
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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EXAMPLE
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For n=3, the 3rd convergent is 161/72 = 2.236111111..., with a sequence of rounded
representations 2, 2.2, 2.24, 2.236, 2.2361, 2.23611, 2.236111, 2.2361111 etc.
Rounded to 1, 2, 3, or 4 post-period decimal digits, this is the same as the rounded version
of the exact sqrt(5), but disagrees if both are rounded to 5 decimal digits, where 2.23607 <> 2.23611.
So a(n=3)= 4 (digits), the maximum rounding level of agreement.
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CROSSREFS
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Cf. A138335, A138336, A138337, A138338, A138339, A138343, A138366, A138369, A138370.
Sequence in context: A026446 A167493 A039121 this_sequence A139143 A047263 A039244
Adjacent sequences: A138364 A138365 A138366 this_sequence A138368 A138369 A138370
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KEYWORD
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nonn,base
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AUTHOR
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Artur Jasinski (grafix(AT)csl.pl), Mar 17 2008
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EXTENSIONS
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Definition and values replaced as defined via continued fractions - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 01 2009
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