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Continued fraction for quaternary expansion of Liouville's number interpreted in base 4 (A012245).
3

%I #23 Nov 21 2019 00:11:33

%S 0,3,5,15,1,4,3,16777215,1,2,4,1,15,5,3,

%T 22300745198530623141535718272648361505980415,1,2,5,15,1,4,2,1,

%U 16777215,3,4,1,15,5,3

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

%H A.H.M. Smeets, <a href="/A317661/b317661.txt">Table of n, a(n) for n = 0..62</a>

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

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

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

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

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

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

%F 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.

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

%o (Python)

%o n,f,i,p,q,base = 1,1,0,0,1,4

%o while i < 100000:

%o ....i,p,q = i+1,p*base,q*base

%o ....if i == f:

%o ........p,n = p+1,n+1

%o ........f = f*n

%o n,a,j = 0,0,0

%o while p%q > 0:

%o ....a,f,p,q = a+1,p//q,q,p%q

%o ....print(a-1,f)

%Y Cf. A012245, A317331, A317332, A317333.

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

%K nonn,base,cofr

%O 0,2

%A _A.H.M. Smeets_, Aug 03 2018