A138374 Count of post-period decimal digits up to which the rounded n-th convergent to 2^(1/3) agrees with the exact value.

1, 1, 2, 2, 3, 4, 4, 6, 6, 8, 6, 10, 10, 12, 13, 15, 16, 17, 16, 18, 19, 20, 21, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 35, 38, 39, 40, 39, 41, 42, 45, 46, 46, 47, 49, 51, 52, 52, 54, 56, 56, 57, 58, 58, 60, 61, 62, 63, 65, 64, 66, 68, 69, 69, 70, 70, 72, 74, 74, 75, 77, 79, 81

Offset

1,3

Comments

This is a measure of the quality of the n-th convergent to the constant A002580 if the convergent and the exact value are compared rounded to an increasing number of digits. The sequence of rounded values of A002580 is 1, 1.3, 1.26, 1.260, 1.2599, 1.25992, 1.259921, 1.2599211 etc. The n-th convergents are taken from A002352 and A002351, 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.

Links

Table of n, a(n) for n=1..73.

Example

For n=5, the 5th convergent is 63/50 = 1.26000000.., with a sequence of rounded representations 1, 1.3, 1.26, 1.260, 1.2600, 1.26000, etc.
Rounded to 1, 2, or 3 post-period decimal digits, this is the same as the rounded version of the exact value, but disagrees if both are rounded to 4 decimal digits, where 1.2599 <> 1.2600.
So a(5) = 3 (digits), the maximum rounding level with agreement.

Crossrefs

Cf. A138335 - A138339, A138343, A138366, A138367, A138369 - A138373.

Sequence in context: A138369 A173332 A348526 * A029936 A114093 A345141

Adjacent sequences: A138371 A138372 A138373 * A138375 A138376 A138377

Keyword

base,nonn

Author

Artur Jasinski, Mar 17 2008

Extensions

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

Status

approved