A138367 Count of post-period decimal digits up to which the rounded n-th convergent to sqrt(5) agrees with the exact value.

0, 2, 4, 5, 6, 7, 8, 10, 8, 12, 14, 14, 16, 18, 19, 20, 21, 23, 24, 24, 26, 28, 29, 30, 31, 33, 33, 34, 35, 37, 39, 40, 41, 42, 44, 44, 46, 47, 48, 49, 51, 53, 53, 55, 56, 57, 59, 60, 60, 61, 64, 65, 66, 68, 69, 70, 72, 73, 74, 75, 76, 77, 79, 80, 81, 83, 83, 85, 85, 88, 89, 90, 91, 92

Offset

1,2

Comments

This is a measure of the quality of the n-th convergent to A002163 if the convergent and the exact value are compared rounded to an increasing number of digits.

The sequence of rounded values of sqrt(5) is 2, 2.2, 2.24, 2.236, 2.2361, 2.23607, 2.236068, 2.2360680 etc, and the n-th convergent (provided by A001077 and A001076) 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..74.

Example

For n=3, the 3rd convergent is 161/72 = 2.236111111..., with a sequence of rounded representations 2, 2.2, 2.24, 2.236, 2.2361, 2.23611, 2.236111, 2.2361111 etc.
Rounded to 1, 2, 3, or 4 post-period decimal digits, this is the same as the rounded version of the exact sqrt(5), but disagrees if both are rounded to 5 decimal digits, where 2.23607 <> 2.23611.
So a(3) = 4 (digits), the maximum rounding level of agreement.

Crossrefs

Cf. A138335, A138336, A138337, A138339, A138343, A138366, A138369, A138370.

Sequence in context: A328107 A368829 A039121 * A360501 A212445 A139143

Adjacent sequences: A138364 A138365 A138366 * A138368 A138369 A138370

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