OFFSET
0,2
COMMENTS
This is a measure of the quality of the n-th convergent to A000796 if the convergent and the exact value are compared rounded to an increasing number of digits. (This is similar to A084407 which compares the truncated/floored values).
The sequence of rounded values of Pi is 3, 3.1, 3.14, 3.142, 3.1416, 3.14159, 3.141593, 3.1415927 etc, and the n-th convergent (provided by A002485 and A002486) 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.
EXAMPLE
For n=3, the 3rd convergent is 355/113 = 3.141592920353.., with a sequence of rounded representations 3, 3.1, 3.14, 3.142, 3.1416, 3.141593, 3.1415929, 3.14159292 etc.
Rounded to 1, 2, 3, 4, 5 or 6 post-period decimal digits, this is the same as the rounded version of the exact Pi, but disagrees if both are rounded to 7 decimal digits, where 3.1415927 <> 3.1415929.
So a(3) = 6 (digits), the maximum rounding level of agreement.
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Artur Jasinski, Mar 16 2008
EXTENSIONS
Definition and values replaced as defined via continued fractions by R. J. Mathar, Oct 01 2009
STATUS
approved