A138372 Count of post-period decimal digits up to which the rounded n-th convergent to A103814 agrees with the exact value.

1, 3, 3, 4, 3, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11, 12, 13, 13, 14, 16, 16, 18, 21, 23, 23, 24, 26, 27, 27, 29, 29, 30, 30, 31, 32, 34, 37, 36, 39, 40, 41, 42, 43, 43, 46, 47, 48, 48, 49, 50, 50, 52, 53, 54, 55, 56, 56, 58, 58, 59, 62, 63, 65, 66, 67, 68, 69, 70, 70, 71, 73, 73, 75

Offset

1,2

Comments

This is a measure of the quality of the n-th convergent to the Pentanacci constant A103814 if the convergent and the exact value are compared rounded to an increasing number of digits. The sequence of rounded values of A103814 is 2, 2.0, 1.97, 1.966, 1.9659, 1.96595, 1.965948, 1.9659482 etc. The n-th convergents are 2 (n=1), 57/29 (n=2), 116/59 (n=3), 173/88 (n=4), 462/235 (n=5) etc, each with associated rounded decimal expansions.

a(n) is the maximum number of post-period digits of the two expansions if compared at the same level of rounding.

Links

Table of n, a(n) for n=1..73.

Example

For n=5, the 5th convergent is 462/235 = 1.96595744.., with a sequence of rounded representations 2, 2.0, 1.97, 1.966, 1.9660, 1.96596, etc.
Rounded to 1, 2, or 3 post-period decimal digits, this is the same as the rounded version of the exact value, but disagrees if both are rounded to 4 decimal digits, where 1.9659 <> 1.9660.
So a(5) = 3 (digits), the maximum rounding level with agreement.

Crossrefs

Cf. A138335 - A138339, A138343, A138366, A138367, A138369 - A138371, A138373 - A138374.

Sequence in context: A096139 A175928 A161357 * A316662 A123708 A350501

Adjacent sequences: A138369 A138370 A138371 * A138373 A138374 A138375

Keyword

base,nonn

Author

Artur Jasinski, Mar 17 2008

Extensions

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

Status

approved