cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A138335 Positions of digits after decimal point in decimal expansion of Pi where the approximation to Pi by a root of a quadratic polynomial does not improve the accuracy.

Original entry on oeis.org

19, 28, 29, 34, 36, 37, 39, 43, 50, 52, 62, 68, 71, 74, 75, 87, 89, 94, 110, 113, 128, 129, 130, 132, 137, 143, 153, 169, 174, 189, 201, 203, 207, 209, 211, 217, 240, 241, 242, 252, 253, 268, 274, 275, 278, 279, 284, 286, 287, 297
Offset: 1

Views

Author

Artur Jasinski, Mar 15 2008

Keywords

Comments

If there is a set of consecutive integers in this sequence starting at k, this means that k-1 is a good approximation to Pi.
If the set of successive integers is longer that approximation k-1 better (see A138336). [Sentence is not clear - N. J. A. Sloane, Dec 09 2017]
Comment from Joerg Arndt, Mar 17 2008: Does Mathematica's N[((quantity)), n] round a number (if so, to what base?) or truncate it? Is Mathematica's Recognize[] guaranteed to give the correct relation? I do not think so: that would be a major breakthrough. That is, this sequence may not even be well-defined.
This sequence is indeed ill defined. One can get the same approximation of Pi to a given precision with infinitely many distinct quadratic polynomials and any such polynomial that gives Pi to n+1 digits also gives Pi to n digits, so this sequence shouldn't have any term. Also, the 18-digit "root" given in the example isn't a root, the polynomial has a value of -5e-13 at this x-value. - M. F. Hasler, May 21 2025

Examples

			a(1)=19 because 3.141592653589793238 (18 digits) is root of -3061495 + 674903*x + 95366*x^2 and 3.1415926535897932385 (19 digits) also is root of that same polynomial -3061495 + 674903*x + 95366*x^2.

Programs

  • Mathematica
    << NumberTheory`Recognize`
b = {}; a = {};
Do[k = Recognize[N[Pi,n], 2, x]; If[MemberQ[a, k], AppendTo[b, n], AppendTo[a, k]], {n, 2, 300}]; b (* Artur Jasinski *)

A138339 Smallest index in A138343 which starts a plateau of n consecutive integers.

Original entry on oeis.org

0, 7, 4, 12
Offset: 1

Views

Author

Artur Jasinski, Mar 16 2008

Keywords

Comments

The sequence provides an indication of regions of nearly no improvements of the sequence of convergents to Pi when measured by the count of correct digits after rounding. A "plateau" of n consecutive integers is defined by a(n) = j such that A138343(j) = A138343(j+n-1) and such that none of the intermediate A138343 with index j+1 up to j+n-2 drop below this level.

Examples

			a(3)=4 because the A138343(4) to A138343(6) are the earliest plateau of length 3.
The fact that there is a local maximum at A138343(5)=9 above the plateau is consistent with the definition.

Crossrefs

Cf. A138335, A138336, A138337, A138343.

Extensions

Definition and values adapted to redefined A138343 - R. J. Mathar, Oct 01 2009

A138343 Count of post-period decimal digits up to which the rounded n-th convergent to Pi agrees with the exact value.

Original entry on oeis.org

0, 2, 3, 6, 8, 9, 8, 10, 10, 11, 11, 13, 15, 15, 16, 15, 17, 17, 18, 19, 20, 23, 24, 23, 26, 27, 29, 30, 29, 31, 33, 34, 37, 39, 39, 40, 42, 43, 44, 45, 45, 47, 46, 49, 49, 51, 52, 52, 54, 55, 56, 55, 56, 57, 59, 58, 59, 60, 61, 61, 63, 64, 64, 65, 65, 66, 67, 67, 68, 69, 70, 71, 72, 72
Offset: 0

Views

Author

Artur Jasinski, Mar 16 2008

Keywords

Comments

This is a measure of the quality of the n-th convergent to A000796 if the convergent and the exact value are compared rounded to an increasing number of digits. (This is similar to A084407 which compares the truncated/floored values).
The sequence of rounded values of Pi is 3, 3.1, 3.14, 3.142, 3.1416, 3.14159, 3.141593, 3.1415927 etc, and the n-th convergent (provided by A002485 and A002486) 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.

Examples

			For n=3, the 3rd convergent is 355/113 = 3.141592920353.., with a sequence of rounded representations 3, 3.1, 3.14, 3.142, 3.1416, 3.141593, 3.1415929, 3.14159292 etc.
Rounded to 1, 2, 3, 4, 5 or 6 post-period decimal digits, this is the same as the rounded version of the exact Pi, but disagrees if both are rounded to 7 decimal digits, where 3.1415927 <> 3.1415929.
So a(3) = 6 (digits), the maximum rounding level of agreement.

Crossrefs

Cf. A138335, A138336, A138337, A138339.

Extensions

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

A138337 Positions of digits after decimal point of number Pi where the approximation to the number Pi by a root of a polynomial of 3 degree does not improve the accuracy.

Original entry on oeis.org

7, 13, 17, 30, 37, 48, 62, 63, 77, 81, 86, 92, 97, 114, 117, 125, 129, 143, 148, 152, 156, 159, 168, 174, 180, 185, 196, 200, 204, 211, 227, 235, 244, 247, 259, 266, 267, 282
Offset: 1

Views

Author

Artur Jasinski, Mar 15 2008

Keywords

Comments

If there is a set of consecutive numbers in this sequence starting at k, this means that k-1 is a good approximation to Pi.
If the set of successive integers is longer that approximation k-1 better (see A138338)
This sequence appears to be ill defined: There are many different polynomials of degree 3 that give an approximation of Pi with the same precision, and any such approximation to n+1 digits is also an approximation of Pi to n digits, so the sequence should be empty. - M. F. Hasler, May 21 2025

Examples

			a(1)=7 because 3.141593 (6 digits) is root of cubic 2 + 29 x - 22 x^2 + 4 x^3 and 3.1415927 (7 digits) also is root of that same polynomial -3061495+674903*x+95366*x^2

Crossrefs

Cf. A138335, A138336, A138338.

Programs

  • Mathematica
    b = {}; a = {}; Do[k = Recognize[N[Pi,n + 1], 3, x]; If[MemberQ[a, k], AppendTo[b, n], AppendTo[a, k]], {n, 2, 300}]; b

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.

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Mar 17 2008

Keywords

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

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

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Mar 17 2008

Keywords

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.

Examples

			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.

Extensions

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

Views

Author

Artur Jasinski, Mar 17 2008

Keywords

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.

Examples

			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.

Extensions

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 4*sin(2*Pi/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

Views

Author

Artur Jasinski, Mar 17 2008

Keywords

Comments

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.

Examples

			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.

Crossrefs

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

Extensions

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

Views

Author

Artur Jasinski, Mar 17 2008

Keywords

Comments

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.

Examples

			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.

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
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