A317661 Continued fraction for quaternary expansion of Liouville's number interpreted in base 4 (A012245).

0, 3, 5, 15, 1, 4, 3, 16777215, 1, 2, 4, 1, 15, 5, 3, 22300745198530623141535718272648361505980415, 1, 2, 5, 15, 1, 4, 2, 1, 16777215, 3, 4, 1, 15, 5, 3

Offset: 0

Author

A.H.M. Smeets, Aug 03 2018

Keywords

• nonn
• base
• cofr

Links

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

Crossrefs

Cf. A012245, A317331, A317332, A317333.

Cf. A058304 (in base 10), A317413 (in base 2), A317414 (in base 3).

Programs

Maple

with(numtheory): cfrac(add(1/4^factorial(n),n=1..7),30,'quotients'); # Muniru A Asiru, Aug 12 2018

Python

n, f, i, p, q, base = 1, 1, 0, 0, 1, 4
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)

Formula

In general for any Liouville's number base > 2:

a(n) = 1 if (and only if, for base > 3) n in A317331,

a(n) = base-2 if (and only if, for base > 3) n in A317332,

a(n) = base-1 if and only if n in A317333,

a(n) = base if and only if n in {8*m - 6 + 3*(m mod 2) | m > 0},

a(n) = base+1 if and only if n in {8*m - 3 - 3*(m mod 2) | m > 0},

a(n) = base^((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.