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 A317413 Continued fraction for binary expansion of Liouville's number interpreted in base 2 (A092874). 6
 0, 1, 3, 3, 1, 2, 1, 4095, 3, 1, 3, 3, 1, 4722366482869645213695, 4, 3, 1, 3, 4095, 1, 2, 1, 3, 3, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The continued fraction of the number obtained by reading A012245 as binary fraction. Except for the first term, the only values that occur in this sequence are 1, 2, 3, 4  and values 2^((m-1)*m!) - 1 for m > 2. The probability of occurrence P(a(n) = k) are given by: P(a(n) = 1) = 1/3, P(a(n) = 2) = 1/12, P(a(n) = 3) = 1/3, P(a(n) = 4) = 1/12 and P(a(n) = 2^((m-1)*m!)-1) = 1/(3*2^(m-1)) for m > 2. The next term is roughly 3.12174855*10^144 (see b-file for precise value). LINKS A.H.M. Smeets, Table of n, a(n) for n = 0..48 FORMULA a(n) = 1 if and only if n in A317538. a(n) = 2 if and only if n in {24*m - 19 | m > 0} union {24*m - 4 | m > 0}. a(n) = 3 if and only if n in A317539. a(n) = 4 if and only if n in {12*m + A014710(m-1) - 2*(A014710(m-1) mod 2) | m > 0} a(n) = 2^((m-1)*m!)-1 if and only if n in {3*2^(m-2)*(1+k*4) - 1 | k >= 0} union {3*2^m-2)*(3+k*4) | k >= 0} for m > 2. EXAMPLE 0.76562505... = 0 +1/( 1+ 1/(3+1/(3+1/(1+1/(2+...))))) - R. J. Mathar, Jun 19 2021 MAPLE with(numtheory): cfrac(add(1/2^factorial(n), n=1..7), 24, 'quotients'); # Muniru A Asiru, Aug 11 2018 MATHEMATICA ContinuedFraction[ FromDigits[ RealDigits[ Sum[1/10^n!, {n, 8}], 10, 10000], 2], 60] (* Robert G. Wilson v, Aug 09 2018 *) PROG (Python) n, f, i, p, q, base = 1, 1, 0, 0, 1, 2 while i < 100000: ....i, p, q = i+1, p*base, q*base ....if i == f: ........p, n = p+1, n+1 ........f = f*n n, a, j = 0, 0, 0 while p%q > 0: ....a, f, p, q = a+1, p//q, q, p%q ....print(a-1, f) CROSSREFS Cf. A012245, A014710, A317538, A317539. Cf. A058304 (in base 10), A317414 (in base 3). Sequence in context: A244328 A073067 A003637 * A110628 A107292 A225331 Adjacent sequences:  A317410 A317411 A317412 * A317414 A317415 A317416 KEYWORD nonn,base,cofr AUTHOR A.H.M. Smeets, Jul 27 2018 STATUS approved

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Last modified November 29 04:38 EST 2021. Contains 349416 sequences. (Running on oeis4.)