A138369 Count of post-period decimal digits up to which the rounded n-th convergent to 4*sin(4*Pi/5) agrees with the exact value.

0, 2, 2, 3, 4, 4, 6, 6, 7, 8, 10, 12, 13, 14, 14, 16, 17, 18, 19, 19, 23, 25, 26, 28, 27, 29, 31, 31, 33, 35, 37, 38, 38, 39, 40, 41, 41, 42, 42, 45, 45, 48, 50, 51, 51, 52, 54, 54, 55, 56, 57, 57, 61, 65, 66, 67, 68, 69, 70, 71, 71, 72, 73, 72, 75, 75, 76, 77, 77, 78, 79, 80, 81, 81, 83

Offset: 2

Author

Artur Jasinski, Mar 17 2008

Keywords

• nonn
• base

Comments

This is a measure of the quality of the n-th convergent to 4*sin(4*Pi/5) = sqrt(2)*sqrt(5-sqrt(5)) = 2.351141009169892... if the convergent and the exact value are compared rounded to an increasing number of digits.

The sequence of rounded values of the sine (or square root) is 2, 2.4, 2.35, 2.351, 2.3511, 2.35114, 2.351141, 2.3511410 etc. The n-th convergents are 5/2 (n=1), 7/3 (n=2), 40/17 (n=3), 47/20, 87/37, 221/94, 308/131 etc. and are represented by their equivalent rounding sequence.

a(n) is the maximum number of post-period digits of the two rounding sequences if compared at the same level of rounding. Counting only post-period digits (which is one less than the total number of decimal digits) is just a convention taken from A084407.

Examples

			For n=4, the 4th convergent is 47/20 = 2.350000000..., with a sequence of rounded representations 2, 2.4, 2.35, 2.350, 2.3500, 2.35000, etc.
Rounded to 1 or 2 post-period decimal digits, this is the same as the rounded version of the exact square root, but disagrees if both are rounded to 3 decimal digits, where 2.351 <> 2.350.
So a(4) = 2 (digits), the maximum rounding level of agreement.
		

Crossrefs

Cf. A138335, A138336, A138337, A138339, A138343, A138366, A138367, A138369, A138370.

Extensions

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