Thue-Morse Constant

On June 20th 2020, I made a post titled Prouhet-Thue-Morse Sequence named for Eugène Prouhet, Axel Thue, and Marston Morse (the Prouhet reference is sometimes omitted). By the way, the Thue part is named after Axel Thue, whose name is pronounced as if it were spelled "Tü" where the ü sound is roughly as in the German word üben. It is incorrect to say "Too-ee" or "Too-eh". Thus sayeth N. J. A. Sloane, June 12th 2018, in his comments about OEIS A010060 that lists the members of the sequence. 

It is a most interesting sequence and my blog post covers it quite well and has links to three interesting YouTube videos. However, there is a so-called Thue-Morse constant that is the topic of this post. The sequence begins:

 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, ... 

If we concatenate these binary digits, we get a binary number:

 \(P=0.0110100110010110100101100..._2\) 

This number can be converted a decimal and is represented by the Greek letter \( \tau \):$$\tau=\sum_{n=0}^{\infty} \frac{t_i}{2^{i+1}}=0.4124540336401075977 \dots$$where \(t_i\) is the \(i^{th}\) element of the binary Thue-Morse sequence. The number has been shown to be transcendental.

Figure 1 provides two interesting expressions for the Thue-Morse constant (source):

Figure 1

I came across the constant by means of my diurnal age investigation, discovering that the number associated with my diurnal age (26547) was a member of OEIS A096394:

A096394

Engel expansion of Thue-Morse constant.

The sequence begins 3, 5, 6, 9, 12, 19, 92, 173, 242, 703, 1861, 3186, 4746, 7843, 26547, ... and the comments state that:$$ 0.4124540336 \dots = \frac{1}{3}+\frac{1}{3 \times 5}+\frac{1}{3 \times 5 \times 6}+\frac{1}{3 \times 5 \times 6 \times 9} + \dots$$I made a post about Engel Expansions way back on September 28th 2016.

If we take 0.412454033640107597783361368258455283089 as an approximation of \(\tau\) and plug this into the SageMathCell formula listed in this post, we do confirm that 26547 is a member. To generate further members of the sequence however, the number of decimal places to which \( \tau \) needs to be approximated must be increased. Here is a permalink to SageMathCell while the code is listed below (blue for input and red for output).

x=0.412454033640107597783361368258455283089

u=x

E=[1]

F=[]

product=1

sum=0

for i in [1..15]:

    a=ceil(1/u)

    u=u*a-1

    E.append(a)

    product=product*a

    sum+=1/product

    F.append(1/product)

print(E, "... this is the Engels expansion")

[1, 3, 5, 6, 9, 12, 19, 92, 173, 242, 703, 1861, 3186, 4746, 7843, 26547] ... this is the Engels expansion