A138366 Count of post-period decimal digits up to which the rounded n-th convergent to exp(1) agrees with the exact value.

0, 1, 0, 1, 2, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 12, 12, 13, 14, 16, 15, 16, 19, 18, 20, 22, 22, 24, 25, 25, 26, 27, 28, 30, 32, 32, 32, 35, 36, 36, 39, 39, 41, 43, 43, 44, 46, 46, 48, 50, 50, 52, 52, 54, 56, 57, 58, 59, 61, 61, 63, 65, 64, 67, 69, 69, 71, 72, 73, 74, 77, 77, 79, 80, 81, 83

Offset

1,5

Comments

This is a measure of the quality of the n-th convergent to E = A001113 if the convergent and the exact value are compared rounded to an increasing number of digits.

The sequence of rounded values of exp(1) is 3, 2.7, 2.72, 2.718, 2.7183, 2.71828, 2.718282, 2.7182818 etc, and the n-th convergent (provided by A007676 and A007677) 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.

Links

Table of n, a(n) for n=1..76.

Example

For n=6, the 6th convergent is 106/39 = 2.7179487.., with a sequence of rounded representations 3, 2.7, 2.72, 2.718, 2.7179, 2.71795, 2.717949, etc.
Rounded to 1, 2, or 3 post-period decimal digits, this is the same as the rounded version of the exact E, but disagrees if both are rounded to 4 decimal digits, where 2.7183 <> 2.7179.
So a(6) = 3 (digits), the maximum rounding level of agreement.

Crossrefs

Cf. A138335, A138336, A138337, A138339, A138343, A138367, A138369, A138370.

Sequence in context: A020892 A196165 A356990 * A065515 A070545 A254828

Adjacent sequences: A138363 A138364 A138365 * A138367 A138368 A138369

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