A084407 -id:A084407 - cp's OEIS frontend

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

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

Artur Jasinski, Mar 16 2008

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.

			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.

Cf. A138335, A138336, A138337, A138339.

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

A138369 Count of post-period decimal digits up to which the rounded n-th convergent to 4sin(4Pi/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

Artur Jasinski, Mar 17 2008

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.

			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.

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

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

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

A305879 Number of binary places to which n-th convergent of continued fraction expansion of Pi matches the correct value.

Original entry on oeis.org

2, 8, 13, 21, 28, 31, 28, 34, 32, 38, 40, 44, 47, 51, 52, 54, 57, 60, 62, 64, 70, 78, 80, 81, 84, 91, 94, 100, 103, 104, 107, 116, 121, 132, 133, 136, 133, 144, 148, 152, 148, 156, 158, 165, 167, 170, 173, 176, 179, 182

Offset: 1

A.H.M. Smeets, Jun 13 2018

For the similar case of number of correct decimal places see A084407.
The denominator of the k-th convergent obtained from a continued fraction satisfying the Gauss-Kuzmin distribution will tend to exp(k*A100199), A100199 being the inverse of Lévy's constant; the error between the k-th convergent and the constant itself tends to exp(-2*k*A100199), or in binary digits 2*k*A100199/log(2) bits after the binary point.
The sequence for quaternary digits is obtained by floor(a(n)/2), the sequence for octal digits is obtained by floor(a(n)/3), the sequence for hexadecimal digits is obtained by floor(a(n)/4).

			Pi = 11.0010010000111111...
n=1: 3/1 = 11.000... so a(1) = 2
n=2: 22/7 = 11.001001001... so a(2) = 8
n=3: 333/106 = 11.00100100001110... so a(3) = 13

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A084407, A086702, A100199, A305607.

Lim {n -> oo} (a(n)/n) = 2*log(A086702)/log(2) = 2*A100199/log(2) = 2*A305607.

A317558 Number of decimal digits to which the n-th convergent of the continued fraction expansion of log(2) matches the correct value.

Original entry on oeis.org

0, -1, 1, 0, 2, 4, 5, 4, 5, 6, 6, 6, 7, 8, 9, 10, 11, 10, 12, 13, 13, 13, 14, 15, 15, 16, 17, 18, 20, 22, 22, 23, 23, 24, 25, 26, 27, 27, 28, 29, 31, 32, 33, 34, 35, 36, 38, 40, 39, 41, 39, 43, 44, 45, 46, 48, 48, 49, 51, 52, 52, 54, 54, 55, 55, 56, 57, 57, 58

Offset: 1

A.H.M. Smeets, Jul 31 2018

Decimal expansion of log(2) in A002162.
For the number of correct binary digits see A317557.
For the similar case of number of correct decimal digits of Pi see A084407.

			   n   convergent    decimal expansion    a(n)
  ==  ============  ====================  ====
   1     0 / 1      0.0                     0
   2     1 / 1      1.0                    -1
   3     2 / 3      0.66...                 1
   4     7 / 10     0.7...                  0
   5     9 / 13     0.692...                2
   6    61 / 88     0.69318...              4
   7   192 / 277    0.693140...             5
   8   253 / 365    0.69315...              4
   9   445 / 642    0.693146...             5
  10  1143 / 1649   0.6931473...            6
  oo  lim = log(2)  0.693147180559945...   --

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A002162, A016730, A086702, A100199, A240995, A317557.

a[n_] := Block[{k = 1, a = RealDigits[Log@2, 10, n + 10][[1]], b = RealDigits[ FromContinuedFraction@ ContinuedFraction[ Log@2, n], 10, n + 10][[1]]}, While[a[[k]] == b[[k]], k++]; k - 1]; a[1] = 0; a[2] = -1; Array[a, 69] (* Robert G. Wilson v, Aug 09 2018 *)

Lim_{n -> oo} a(n)/n = 2*log(A086702)/log(10) = 2*A100199/log(10) = 2*A240995.

a(61) onward from Robert G. Wilson v, Aug 09 2018

A317908 Number of decimal places to which the n-th convergent of the continued fraction expansion of Khintchine's constant matches the correct value.

Original entry on oeis.org

0, -1, 1, 2, 2, 3, 3, 4, 4, 6, 5, 8, 8, 9, 11, 13, 12, 14, 15, 16, 16, 16, 18, 21, 21, 23, 24, 24, 25, 25, 26, 27, 28, 29, 30, 30, 32, 32, 33, 33, 36, 35, 36, 37, 37, 38, 39, 39, 40, 41, 42, 42, 43, 44, 45, 44, 46, 47, 48, 48, 49, 50, 51, 54, 55, 56, 56, 58, 58, 60

Offset: 1

A.H.M. Smeets, Aug 10 2018

Decimal expansion of Khintchine's constant in A002210.
For the similar case of the number of correct decimal digits of Pi see A084407.
For the similar case of the number of correct decimal digits of log(2) see A317558.
For the number of correct binary places see A317907.

			   n   convergent     decimal expansion    a(n)
  ==  =============  ====================  ====
   1     2 / 1       2.0                     0
   2     3 / 1       3.0                    -1
   3     8 / 3       2.66...                 1
   4    43 / 16      2.687...                2
   5    51 / 19      2.684...                2
   6    94 / 35      2.6857...               3
   7   239 / 89      2.6853...               3
   8   333 / 124     2.68548...              4
   9   572 / 213     2.68544...              4
  10  2049 / 763     2.6854521...            6
  oo  lim = A002210  2.685452001065306...   --

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A002210, A002211, A086702, A100199, A240995, A317907.

i,cf = 0,[]
while i <= 20100:
    c = A002211(i)
    cf,i = cf+[c],i+1
p0,p1,q0,q1,i,base = cf[0],1,1,0,1,10
while i <= 20100:
    p0,p1,q0,q1,i = cf[i]*p0+p1,p0,cf[i]*q0+q1,q0,i+1
a0 = p0//q0
p0 = p0-a0*q0
i,p0,dd = 0,p0*base,[a0]
while i < 21000:
    d,p0,i = p0//q0,(p0%q0)*base,i+1
    dd = dd+[d]
n,pn0,pn1,qn0,qn1 = 1,a0,1,1,0
while n <= 20000:
    p,q = pn0,qn0
    if p//q != a0:
        print(n,"- manual!")
    else:
        i,p,di = 0,(p%q)*base,a0
        while di == dd[i]:
            i,di,p = i+1,p//q,(p%q)*base
        print(n,i-1)
    n,pn0,pn1,qn0,qn1 = n+1,cf[n]*pn0+pn1,pn0,cf[n]*qn0+qn1,qn0

Limit_{n -> oo} (a(n)/n) = 2*log(A086702)/log(10) = 2*A100199/log(10) = 2*A240995.

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

A236250 Period of the n-th convergent to the continued fraction expansion of Pi.

Original entry on oeis.org

1, 6, 13, 112, 51, 24, 15088, 12284, 88460, 1204, 459, 31824, 93210, 1864254, 531648, 456036, 8299090, 28574910, 1813560, 32552820, 33166008, 133585180, 2503410, 214098720, 3183870690, 7411133309730, 4852769490690, 2294509753536, 175964053944, 3336533898768

Offset: 1

Jani Melik, Jan 21 2014

			The 2nd convergent is 22/7 = 3.142857 142857 ..., whose period is 6, so a(2) = 6.
The 3rd convergent is 333/106 = 3.1 4150943396226 4150943396226 ..., whose period is 13, so a(3) = 13.

Cf. A001203, A007732, A084407.

st_clenov = 30
def A236250(n) :
   vu = continued_fraction_list(pi, nterms=st_clenov);
   p = []
   for i in (0..n) :
      p.append(convergents(vu)[i].period())
   return(p)
A236250(st_clenov-1);

a(n) = A007732(A002486(n+2)). - Michel Marcus, Jan 21 2014

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

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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.

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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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A305879 Number of binary places to which n-th convergent of continued fraction expansion of Pi matches the correct value.

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A317558 Number of decimal digits to which the n-th convergent of the continued fraction expansion of log(2) matches the correct value.

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Mathematica

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A317908 Number of decimal places to which the n-th convergent of the continued fraction expansion of Khintchine's constant matches the correct 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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A236250 Period of the n-th convergent to the continued fraction expansion of Pi.

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