0,2

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.

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(n=3)= 6 (digits), the maximum rounding level of agreement.

Cf. A138335, A138336, A138337, A138338, A138339.

Sequence in context: A140266 A140265 A127293 this_sequence A139371 A079338 A047405

Adjacent sequences: A138340 A138341 A138342 this_sequence A138344 A138345 A138346

nonn,base

Artur Jasinski (grafix(AT)csl.pl), Mar 16 2008

Definition and values replaced as defined via continued fractions - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 01 2009