A359457 Continued fraction for constant A359456.

0, 9, 11, 99, 1, 10, 9, 999999999999999999, 1, 8, 10, 1, 99, 11, 9

Offset: 0

Author

A.H.M. Smeets, Jan 02 2023

Keywords

• nonn
• cofr
• more

Comments

It seems that all terms smaller than 100 are identical to the continued fraction terms of Liouville's constant as in A058304.

Except for the first term, the only values that occur in this sequence are 1,8,9,10,11,and values 10^A359458(m-1)-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^A359458(m-1)-1) = 2^-(m+1) for m > 1.

Crossrefs

Cf. A058304, A317331, A317332, A317333, A359456, A359458.

Formula

a(n) = 1 if and only if n in A317331,

a(n) = 8 if and only if n in A317332,

a(n) = 9 if and only if n in A317333,

a(n) = 10 if and only if n = 8*m - 6 + 3*(m mod 2) for m > 0,

a(n) = 11 if and only if n = 8*m - 3 - 3*(m mod 2) for m > 0,

a(n) = 10^A359458(m-1)-1 if and only if n in {2^m*(1+k*4) - 1 | k >= 0} union {2^m*(3+k*4) | k >= 0} for m > 1.