

A058304


Continued fraction for Liouville's number (A012245).


8



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
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OFFSET

0,2


COMMENTS

From A.H.M. Smeets, Jun 06 2018: (Start)
Except for the first term, the only values that occur in this sequence are 1,8,9,10,11,and values 10^((m1)*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^((m1)*m!)1) = 2^(m+1) for m > 1. (End)


REFERENCES

Harold M. Stark, "An Introduction to Number Theory," The MIT Press, Cambridge, MA and London, England, Eighth Printing, 1994, pages 172  177.


LINKS

Muniru A Asiru, Table of n, a(n) for n = 0..62
J. O. Shallit, Simple Continued Fractions for Some Irrational Numbers II, J. Number Theory 14 (1982), 228231.
Eric Weisstein's World of Mathematics, Liouville's Constant
G. Xiao, Contfrac
Index entries for continued fractions for constants


FORMULA

From A.H.M. Smeets, Jun 26 2018: (Start)
a(n) = 1 iff n in A317331,
a(n) = 8 iff n in A317332,
a(n) = 9 iff n in A317333,
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^((m1)*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)


EXAMPLE

0.1100010000000000000000010... = 0 + 1/(9 + 1/(11 + 1/(99 + 1/(1 + ...)))).  Harry J. Smith, May 15 2009


MAPLE

with(numtheory): cfrac(add(1/10^factorial(n), n=1..7), 62, 'quotients'); # Muniru A Asiru, Aug 08 2018


MATHEMATICA

ContinuedFraction[ Sum[ 1 /10^(n!), {n, 1, 7} ], 40 ]


PROG

(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(a1, f)
# A.H.M. Smeets, Aug 03 2018


CROSSREFS

Cf. A012245.
Cf. A317413 (in base 2), A317414 (in base 3) A317661 (in base 4 and general).
Sequence in context: A137016 A053886 A137020 * A027727 A019328 A119786
Adjacent sequences: A058301 A058302 A058303 * A058305 A058306 A058307


KEYWORD

cofr,nonn


AUTHOR

Robert G. Wilson v, Dec 08 2000


EXTENSIONS

Offset changed to 0 on the advice of A.H.M. Smeets by Muniru A Asiru, Aug 11 2018


STATUS

approved



