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A058304
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Continued fraction for Liouville's number (A012245).
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9
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0, 9, 11, 99, 1, 10, 9, 999999999999, 1, 8, 10, 1, 99, 11, 9, 999999999999999999999999999999999999999999999999999999999999999999999999, 1, 8, 11, 99, 1, 10, 8, 1, 999999999999, 9, 10, 1, 99, 11, 9
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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Except for the first term, the only values that occur in this sequence are 1,8,9,10,11,and values 10^((m-1)*m!)-1 for m > 1. The probability of occurrence P(a(n) = k) are given by:
P(a(n) = 1) = 1/4,
P(a(n) = 8) = 1/8,
P(a(n) = 9) = 1/8,
P(a(n) = 10) = 1/8,
P(a(n) = 11) = 1/8 and
P(a(n) = 10^((m-1)*m!)-1) = 2^-(m+1) for m > 1. (End)
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REFERENCES
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Harold M. Stark, "An Introduction to Number Theory," The MIT Press, Cambridge, MA and London, England, Eighth Printing, 1994, pages 172 - 177.
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LINKS
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FORMULA
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a(n) = 10 iff n = 8*m - 6 + 3*(m mod 2) for m > 0,
a(n) = 11 iff n = 8*m - 3 - 3*(m mod 2) for m > 0,
a(n) = 10^((m-1)*m!)-1 iff n in {2^m*(1+k*4) - 1 | k >= 0} union {2^m*(3+k*4) | k >= 0} for m > 1. (End)
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EXAMPLE
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0.1100010000000000000000010... = 0 + 1/(9 + 1/(11 + 1/(99 + 1/(1 + ...)))). - Harry J. Smith, May 15 2009
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MAPLE
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with(numtheory): cfrac(add(1/10^factorial(n), n=1..7), 62, 'quotients'); # Muniru A Asiru, Aug 08 2018
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MATHEMATICA
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ContinuedFraction[ Sum[ 1 /10^(n!), {n, 1, 7} ], 40 ]
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PROG
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(PARI) { allocatemem(932245000); default(realprecision, 200000); x=contfrac(suminf(n=1, 1.0/10^n!)); for (n=1, 255, write("b058304.txt", n, " ", x[n])); } \\ Harry J. Smith, May 15 2009
(Python)
n, f, i, p, q, base = 1, 1, 0, 0, 1, 10
while i < 1000:
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)
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CROSSREFS
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KEYWORD
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cofr,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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