A138369 -id:A138369 - cp's OEIS frontend

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

Artur Jasinski, Mar 17 2008

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.

			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.

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

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

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

Original entry on oeis.org

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

Artur Jasinski, Mar 17 2008

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.

			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.

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

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

A138370 Count of post-period decimal digits up to which the rounded n-th convergent to 4sin(2Pi/5) agrees with the exact value.

Original entry on oeis.org

2, 3, 4, 5, 5, 6, 6, 8, 9, 10, 11, 12, 12, 13, 14, 15, 15, 17, 18, 17, 19, 21, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 33, 34, 35, 36, 37, 38, 38, 40, 41, 42, 41, 42, 43, 45, 44, 46, 44, 47, 49, 49, 50, 52, 53, 54, 55, 57, 59, 60, 61, 63, 62, 65, 67, 67, 68, 70, 69, 70, 70, 71

Offset: 2

Artur Jasinski, Mar 17 2008

The computation of A138369 is repeated for 4*sin(2*Pi/5) = sqrt(2)*sqrt(5+sqrt(5))
= 3.80422606518061.. = 4*A019881.
The convergents are 19/5 (n=2), 175/46 (n=3), 544/143 (n=4), 719/189 (n=5), 2701/710 (n=6) etc.

			a(6)=5 because 2701/710 = 3.80422535... agrees with 3.8042260651.. if both are rounded up to 5 decimal digits (3.80423 = 3.80423), but disagrees at the level of rounding to 6 decimal digits (3.804226 <> 3.804225) or more.

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

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

A138373 Count of post-period decimal digits up to which the rounded n-th convergent to sqrt(5)/2 agrees with the exact value.

Original entry on oeis.org

1, 3, 3, 5, 6, 8, 9, 10, 9, 13, 13, 15, 15, 18, 19, 20, 22, 22, 24, 25, 26, 27, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 43, 44, 45, 47, 47, 49, 50, 52, 53, 54, 55, 56, 58, 57, 60, 61, 62, 64, 64, 67, 68, 68, 71, 72, 73, 74, 75, 76, 78, 78, 80, 82, 83, 84, 85, 86, 88, 88, 90

Offset: 1

Artur Jasinski, Mar 17 2008

Defined equivalent to A138367 considering the constant 1.1188.. = 10*A020837 and its convergents 9/8 (n=1), 19/17 (n=2), 161/144 (n=3), 341/305 (n=4), 2889/2584 (n=5).

Cf. A138335 - A138339, A138343, A138366, A138367, A138369, A138370 - A138374.

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

A138374 Count of post-period decimal digits up to which the rounded n-th convergent to 2^(1/3) agrees with the exact value.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 6, 6, 8, 6, 10, 10, 12, 13, 15, 16, 17, 16, 18, 19, 20, 21, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 35, 38, 39, 40, 39, 41, 42, 45, 46, 46, 47, 49, 51, 52, 52, 54, 56, 56, 57, 58, 58, 60, 61, 62, 63, 65, 64, 66, 68, 69, 69, 70, 70, 72, 74, 74, 75, 77, 79, 81

Offset: 1

Artur Jasinski, Mar 17 2008

This is a measure of the quality of the n-th convergent to the constant A002580 if the convergent and the exact value are compared rounded to an increasing number of digits. The sequence of rounded values of A002580 is 1, 1.3, 1.26, 1.260, 1.2599, 1.25992, 1.259921, 1.2599211 etc. The n-th convergents are taken from A002352 and A002351, 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.

			For n=5, the 5th convergent is 63/50 = 1.26000000.., with a sequence of rounded representations 1, 1.3, 1.26, 1.260, 1.2600, 1.26000, 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 1.2599 <> 1.2600.
So a(5) = 3 (digits), the maximum rounding level with agreement.

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

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

A138371 Count of post-period decimal digits up to which the rounded n-th convergent to A058265 agrees with the exact value.

Original entry on oeis.org

0, 1, 2, 5, 8, 7, 10, 11, 10, 12, 15, 17, 17, 17, 20, 21, 22, 23, 25, 26, 28, 30, 29, 30, 31, 31, 32, 32, 34, 35, 35, 36, 36, 38, 40, 40, 42, 42, 42, 43, 44, 43, 45, 46, 47, 47, 49, 52, 51, 52, 54, 54, 55, 57, 59, 59, 60, 60, 60, 61, 61, 62, 62, 64, 64, 66, 67, 69, 71, 73, 74

Offset: 1

Artur Jasinski, Mar 17 2008

This is a measure of the quality of the n-th convergent to the tribonacci constant A058265 if the convergent and the exact value are compared rounded to an increasing number of digits. The sequence of rounded values of A058265 is 2, 1.8, 1.84, 1.839, 1.8393, 1.83929, 1.839287, 1.8392868, etc. The n-th convergents are 2 (n=1), 11/6 (n=2), 46/25 (n=3), 103/56 (n=4), 31451/17105 (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. 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=4, the 4th convergent is 103/56 = 1.83928571..., with a sequence of rounded representations 2, 1.8, 1.84, 1.839, 1.8393, 1.83929, 1.839286, 1.8392857 etc.
Rounded to 1, 2, 3, 4 or 5 post-period decimal digits, this is the same as the rounded version of the exact value, but disagrees if both are rounded to 6 decimal digits, where 1.839287 <> 1.839286.
So a(4) = 5 (digits), the maximum rounding level with agreement.

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

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

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

Original entry on oeis.org

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

Artur Jasinski, Mar 17 2008

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.

			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.

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

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

cp's OEIS Frontend

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

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

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A138370 Count of post-period decimal digits up to which the rounded n-th convergent to 4*sin(2*Pi/5) agrees with the exact value.

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A138373 Count of post-period decimal digits up to which the rounded n-th convergent to sqrt(5)/2 agrees with the exact value.

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A138374 Count of post-period decimal digits up to which the rounded n-th convergent to 2^(1/3) agrees with the exact value.

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A138371 Count of post-period decimal digits up to which the rounded n-th convergent to A058265 agrees with the exact value.

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Examples

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

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Examples

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Extensions

A138370 Count of post-period decimal digits up to which the rounded n-th convergent to 4sin(2Pi/5) agrees with the exact value.