A086702 -id:A086702 - cp's OEIS frontend

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

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.

A247039 Decimal expansion of the analog of Lévy's constant in case of the nearest integer continued fraction of -1/2

Original entry on oeis.org

5, 5, 2, 4, 3, 0, 7, 9, 7, 0, 2, 5, 0, 9, 2, 8, 9, 6, 0, 8, 6, 2, 7, 0, 5, 8, 7, 5, 4, 4, 0, 8, 2, 9, 3, 7, 7, 5, 6, 3, 1, 0, 2, 9, 0, 3, 3, 2, 8, 4, 1, 9, 2, 3, 7, 0, 3, 4, 7, 0, 2, 5, 0, 5, 5, 7, 9, 7, 8, 2, 3, 6, 5, 8, 1, 3, 2, 0, 0, 3, 8, 5, 7, 7, 4, 3, 5, 8, 4, 8, 0, 2, 9, 6, 7, 7, 5, 7, 9, 6

Offset: 1

Jean-François Alcover, Sep 10 2014

			5.524307970250928960862705875440829377563102903328419237...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.8 Khinchin-Lévy constants p. 62.

Eric Weisstein's MathWorld, Lévy's Constant

Cf. A086702.

RealDigits[Exp[Pi^2/(12*Log[GoldenRatio])], 10, 100] // First

exp(Pi^2/(12*log(phi))), where phi is the golden ratio (1+sqrt(5))/2.

A275696 Decimal expansion of e^(3Zeta(3)/(4log(2))).

Original entry on oeis.org

3, 6, 7, 1, 6, 8, 6, 7, 0, 7, 4, 3, 0, 0, 6, 4, 5, 0, 0, 7, 8, 0, 6, 1, 4, 1, 4, 9, 9, 0, 9, 9, 7, 8, 2, 7, 4, 6, 6, 1, 5, 9, 3, 1, 8, 3, 5, 4, 8, 9, 6, 8, 2, 7, 0, 3, 4, 6, 8, 0, 0, 1, 1, 9, 7, 0, 5, 2, 1, 6, 5, 6, 6, 8, 9, 8, 3, 4, 8, 0, 0, 3, 6, 1, 5, 7, 3, 6, 6, 2, 5, 0, 1, 4, 5, 1, 1, 6, 2, 3

Offset: 1

Terry D. Grant, Aug 05 2016

			3.67168670743006450078...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A275689, A086702, A089729, A100199.

RealDigits[Exp[3*(Zeta[3]/(4*Log[2]))],10,100]

exp(3*(zeta(3)/(4*log(2)))) \\ G. C. Greubel, Mar 23 2017

Equals e^(3*Zeta(3)/(4*log(2))) = A001113^(A275689).

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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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A247039 Decimal expansion of the analog of Lévy's constant in case of the nearest integer continued fraction of -1/2

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A275696 Decimal expansion of e^(3Zeta(3)/(4log(2))).

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cp's OEIS Frontend

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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A247039 Decimal expansion of the analog of Lévy's constant in case of the nearest integer continued fraction of -1/2

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Mathematica

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A275696 Decimal expansion of e^(3*Zeta(3)/(4*log(2))).

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A275696 Decimal expansion of e^(3Zeta(3)/(4log(2))).