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Number of decimal places to which the n-th convergent of the continued fraction expansion of Khintchine's constant matches the correct value.
2

%I #18 Jan 27 2022 21:59:44

%S 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,

%T 24,25,25,26,27,28,29,30,30,32,32,33,33,36,35,36,37,37,38,39,39,40,41,

%U 42,42,43,44,45,44,46,47,48,48,49,50,51,54,55,56,56,58,58,60

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

%C Decimal expansion of Khintchine's constant in A002210.

%C For the similar case of the number of correct decimal digits of Pi see A084407.

%C For the similar case of the number of correct decimal digits of log(2) see A317558.

%C For the number of correct binary places see A317907.

%H A.H.M. Smeets, <a href="/A317908/b317908.txt">Table of n, a(n) for n = 1..20000</a>

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

%e n convergent decimal expansion a(n)

%e == ============= ==================== ====

%e 1 2 / 1 2.0 0

%e 2 3 / 1 3.0 -1

%e 3 8 / 3 2.66... 1

%e 4 43 / 16 2.687... 2

%e 5 51 / 19 2.684... 2

%e 6 94 / 35 2.6857... 3

%e 7 239 / 89 2.6853... 3

%e 8 333 / 124 2.68548... 4

%e 9 572 / 213 2.68544... 4

%e 10 2049 / 763 2.6854521... 6

%e oo lim = A002210 2.685452001065306... --

%o (Python)

%o i,cf = 0,[]

%o while i <= 20100:

%o ....c = A002211(i)

%o ....cf,i = cf+[c],i+1

%o p0,p1,q0,q1,i,base = cf[0],1,1,0,1,10

%o while i <= 20100:

%o ....p0,p1,q0,q1,i = cf[i]*p0+p1,p0,cf[i]*q0+q1,q0,i+1

%o a0 = p0//q0

%o p0 = p0-a0*q0

%o i,p0,dd = 0,p0*base,[a0]

%o while i < 21000:

%o ....d,p0,i = p0//q0,(p0%q0)*base,i+1

%o ....dd = dd+[d]

%o n,pn0,pn1,qn0,qn1 = 1,a0,1,1,0

%o while n <= 20000:

%o ....p,q = pn0,qn0

%o ....if p//q != a0:

%o ........print(n,"- manual!")

%o ....else:

%o ........i,p,di = 0,(p%q)*base,a0

%o ........while di == dd[i]:

%o ............i,di,p = i+1,p//q,(p%q)*base

%o ........print(n,i-1)

%o ....n,pn0,pn1,qn0,qn1 = n+1,cf[n]*pn0+pn1,pn0,cf[n]*qn0+qn1,qn0

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

%K sign,base

%O 1,4

%A _A.H.M. Smeets_, Aug 10 2018