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.
%I A317907 #23 May 22 2025 10:21:48
%S A317907 0,-1,5,3,9,8,12,14,16,22,25,27,30,33,39,44,42,49,52,51,56,55,64,70,
%T A317907 73,77,81,83,82,85,88,92,93,99,101,104,109,104,111,114,117,120,122,
%U A317907 124,126,129,131,133,136,139,138,144,138,148,151,150,153,156,158,162
%N A317907 Number of binary places to which n-th convergent of continued fraction expansion of Khintchine's constant matches the correct value.
%C A317907 For number of correct decimal digits see A317908.
%C A317907 For the similar case of number of correct binary digits of Pi see A305879.
%C A317907 For the similar case of number of correct binary digits of log(2) see A317557.
%C A317907 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.
%C A317907 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).
%H A317907 A.H.M. Smeets, <a href="/A317907/b317907.txt">Table of n, a(n) for n = 1..19999</a>
%F A317907 Lim_{n -> oo} a(n)/n = 2*log(A086702)/log(2) = 2*A100199/log(2) = 2*A305607.
%e A317907 n convergent binary expansion a(n)
%e A317907 == ============= ========================== ====
%e A317907 1 2 / 1 10.0... 0
%e A317907 2 3 / 1 11.0... -1
%e A317907 3 8 / 3 10.101010... 5
%e A317907 4 43 / 16 10.1011... 3
%e A317907 5 51 / 19 10.1010111100... 9
%e A317907 oo lim = A317906 10.101011110111100111... --
%o A317907 (Python)
%o A317907 i,cf = 0,[]
%o A317907 while i <= 20100:
%o A317907 c = A002211(i)
%o A317907 cf,i = cf+[c],i+1
%o A317907 p0,p1,q0,q1,i,base = cf[0],1,1,0,1,2
%o A317907 while i <= 20100:
%o A317907 p0,p1,q0,q1,i = cf[i]*p0+p1,p0,cf[i]*q0+q1,q0,i+1
%o A317907 a0 = p0//q0
%o A317907 p0 = p0-a0*q0
%o A317907 i,p0,dd = 0,p0*base,[a0]
%o A317907 while i < 70000:
%o A317907 d,p0,i = p0//q0,(p0%q0)*base,i+1
%o A317907 dd = dd+[d]
%o A317907 n,pn0,pn1,qn0,qn1 = 1,a0,1,1,0
%o A317907 while n <= 20000:
%o A317907 p,q = pn0,qn0
%o A317907 if p//q != a0:
%o A317907 print(n,"- manual!")
%o A317907 else:
%o A317907 i,p,di = 0,(p%q)*base,a0
%o A317907 while di == dd[i]:
%o A317907 i,di,p = i+1,p//q,(p%q)*base
%o A317907 print(n,i-1)
%o A317907 n,pn0,pn1,qn0,qn1 = n+1,cf[n]*pn0+pn1,pn0,cf[n]*qn0+qn1,qn0
%Y A317907 Cf. A002210, A002211, A086702, A100199, A305607, A317906, A317908.
%K A317907 sign,base
%O A317907 1,3
%A A317907 _A.H.M. Smeets_, Aug 10 2018