

A138369


Count of postperiod decimal digits up to which the rounded nth convergent to 4*sin(4*Pi/5) agrees with the exact value.


8



0, 2, 2, 3, 4, 4, 6, 6, 7, 8, 10, 12, 13, 14, 14, 16, 17, 18, 19, 19, 23, 25, 26, 28, 27, 29, 31, 31, 33, 35, 37, 38, 38, 39, 40, 41, 41, 42, 42, 45, 45, 48, 50, 51, 51, 52, 54, 54, 55, 56, 57, 57, 61, 65, 66, 67, 68, 69, 70, 71, 71, 72, 73, 72, 75, 75, 76, 77, 77, 78, 79, 80, 81, 81, 83
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OFFSET

2,2


COMMENTS

This is a measure of the quality of the nth convergent to 4*sin(4*Pi/5) = sqrt(2)*sqrt(5sqrt(5)) = 2.351141009169892... if the convergent and the exact value are compared rounded to an increasing number of digits.
The sequence of rounded values of the sine (or square root) is 2, 2.4, 2.35, 2.351, 2.3511, 2.35114, 2.351141, 2.3511410 etc. The nth convergents are 5/2 (n=1), 7/3 (n=2), 40/17 (n=3), 47/20, 87/37, 221/94, 308/131 etc. and are represented by their equivalent rounding sequence.
a(n) is the maximum number of postperiod digits of the two rounding sequences if compared at the same level of rounding. Counting only postperiod digits (which is one less than the total number of decimal digits) is just a convention taken from A084407.


LINKS

Table of n, a(n) for n=2..76.


EXAMPLE

For n=4, the 4th convergent is 47/20 = 2.350000000..., with a sequence of rounded representations 2, 2.4, 2.35, 2.350, 2.3500, 2.35000, etc.
Rounded to 1 or 2 postperiod decimal digits, this is the same as the rounded version of the exact square root, but disagrees if both are rounded to 3 decimal digits, where 2.351 <> 2.350.
So a(4) = 2 (digits), the maximum rounding level of agreement.


CROSSREFS

Cf. A138335, A138336, A138337, A138339, A138343, A138366, A138367, A138369, A138370.
Sequence in context: A126246 A338517 A173633 * A173332 A348526 A138374
Adjacent sequences: A138366 A138367 A138368 * A138370 A138371 A138372


KEYWORD

nonn,base


AUTHOR

Artur Jasinski, Mar 17 2008


EXTENSIONS

Definition and values replaced as defined via continued fractions by R. J. Mathar, Oct 01 2009


STATUS

approved



