A305879 -id:A305879 - cp's OEIS frontend

A317557 Number of binary digits to which the n-th convergent of the continued fraction expansion of log(2) matches the correct value.

0, -1, 3, 6, 9, 13, 14, 17, 19, 20, 23, 20, 25, 20, 33, 37, 35, 38, 41, 43, 45, 43, 47, 48, 52, 54, 58, 61, 68, 70, 74, 77, 78, 81, 86, 89, 92, 93, 92, 99, 105, 109, 113, 116, 118, 121, 127, 133, 136, 135, 139, 141, 145, 149, 154, 159, 161, 165, 171, 173, 172, 180

Offset: 1

A.H.M. Smeets, Jul 31 2018

Binary expansion of log(2) in A068426.
For number of correct decimal digits see A317558.
For the similar case of number of correct binary digits of Pi see A305879.
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), and the sequence for hexadecimal digits is obtained by floor(a(n)/4).

			   n   convergent         binary expansion         a(n)
  ==  ============  =============================  ====
   1     0 / 1      0.0                              0
   2     1 / 1      1.0                             -1
   3     2 / 3      0.1010...                        3
   4     7 / 10     0.1011001...                     6
   5     9 / 13     0.1011000100...                  9
   6    61 / 88     0.10110001011101...             13
   7   192 / 277    0.101100010111000...            14
   8   253 / 365    0.101100010111001001...         17
   9   445 / 642    0.10110001011100100000...       19
  10  1143 / 1649   0.101100010111001000011...      20
  oo  lim = log(2)  0.101100010111001000010111...   --

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

Cf. A016730, A068426, A086702, A100199, A305607, A317558.

a[n_] := Block[{k = 1, a = RealDigits[ Log@2, 2, 4 + 10][[1]], b = RealDigits[ FromContinuedFraction@ ContinuedFraction[Log@2, n + 1], 2, 4n + 10][[1]]}, While[ a[[k]] == b[[k]], k++]; k - 1]; a[1] = 0; a[2] = -1; Array[a, 61] (* Robert G. Wilson v, Aug 09 2018 *)

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

a(40) onward from Robert G. Wilson v, Aug 09 2018

A317907 Number of binary places to which n-th convergent of continued fraction expansion of Khintchine's constant matches the correct value.

Original entry on oeis.org

0, -1, 5, 3, 9, 8, 12, 14, 16, 22, 25, 27, 30, 33, 39, 44, 42, 49, 52, 51, 56, 55, 64, 70, 73, 77, 81, 83, 82, 85, 88, 92, 93, 99, 101, 104, 109, 104, 111, 114, 117, 120, 122, 124, 126, 129, 131, 133, 136, 139, 138, 144, 138, 148, 151, 150, 153, 156, 158, 162

Offset: 1

A.H.M. Smeets, Aug 10 2018

For number of correct decimal digits see A317908.
For the similar case of number of correct binary digits of Pi see A305879.
For the similar case of number of correct binary digits of log(2) see A317557.
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), and the sequence for hexadecimal digits is obtained by floor(a(n)/4).

			   n   convergent         binary expansion       a(n)
  ==  =============  ==========================  ====
   1     2 / 1       10.0...                       0
   2     3 / 1       11.0...                      -1
   3     8 / 3       10.101010...                  5
   4    43 / 16      10.1011...                    3
   5    51 / 19      10.1010111100...              9
  oo  lim = A317906  10.101011110111100111...     --

A.H.M. Smeets, Table of n, a(n) for n = 1..19999

Cf. A002210, A002211, A086702, A100199, A305607, A317906, A317908.

i,cf = 0,[]
while i <= 20100:
    c = A002211(i)
    cf,i = cf+[c],i+1
p0,p1,q0,q1,i,base = cf[0],1,1,0,1,2
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 < 70000:
    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

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

cp's OEIS Frontend

A317557 Number of binary digits to which the n-th convergent of the continued fraction expansion of log(2) matches the correct value.

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