A317661 -id:A317661 - cp's OEIS frontend

A058304 Continued fraction for Liouville's number (A012245).

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

Offset: 0

Robert G. Wilson v, Dec 08 2000

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^((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)

			0.1100010000000000000000010... = 0 + 1/(9 + 1/(11 + 1/(99 + 1/(1 + ...)))). - _Harry J. Smith_, May 15 2009

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

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), 228-231.
Eric Weisstein's World of Mathematics, Liouville's Constant
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A012245.

Cf. A317413 (in base 2), A317414 (in base 3) A317661 (in base 4 and general).

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

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

{ 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

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)
# A.H.M. Smeets, Aug 03 2018

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^((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)

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

A317414 Continued fraction for ternary expansion of Liouville's number interpreted in base 3 (A012245).

Original entry on oeis.org

0, 2, 4, 8, 1, 3, 2, 531440, 1, 1, 3, 1, 8, 4, 2, 22528399544939174411840147874772640, 1, 1, 4, 8, 1, 3, 1, 1, 531440, 2, 3, 1, 8, 4, 2

Offset: 0

A.H.M. Smeets, Jul 27 2018

The continued fraction of the number obtained by reading A012245 as a ternary fraction.
Except for the first term, the only values that occur in this sequence are 1,2,3,4, and values 3^((m-1)*m!)-1 for m > 1. The probability of occurrence P(a(n) = k) are given by:
P(a(n) = 1) = 3/8,
P(a(n) = 2) = 1/8,
P(a(n) = 3) = 1/8,
P(a(n) = 4) = 1/8 and
P(a(n) = 3^((m-1)*m!)-1) = 2^-(m+1) for m > 1.
More generally it seems that for any base > 2, P(a(n) <= base+1) = 3/4, P(a(n) > base+1) = 1/4, and P(a(n) = base^((m-1)*m!)-1) = 2^-(m+1) for m > 1.

A.H.M. Smeets, Table of n, a(n) for n = 0..62

Cf. A012245, A089013, A317627.

Cf. A058304 (in base 10), A317413 (in base 2), A317661 (in base 4).

with(numtheory): cfrac(add(1/3^factorial(n),n=1..7),30,'quotients'); # Muniru A Asiru, Aug 11 2018

ContinuedFraction[ FromDigits[ RealDigits[ Sum[1/10^n!, {n, 8}], 10, 10000], 3], 60] (* Robert G. Wilson v, Aug 09 2018 *)

n,f,i,p,q,base = 1,1,0,0,1,3
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)

a(n) = 1 if and only if n in {floor(8*n/3) + A317627(n) | n > 0}.
a(n) = 2 if and only if n in {8*n - 10 + 3*A089013(n-1) | n > 0}.
a(n) = 3 if and only if n in {16*n - 11 | n > 0} union {16*n - 6 | n > 0}.
a(n) = 4 if and only if n in {16*n - 14 | n > 0} union {16*n - 3 | n > 0}.
a(n) = 3^((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.

cp's OEIS Frontend

A058304 Continued fraction for Liouville's number (A012245).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions