# Number defined by a recursive binary sequence

In a math column in Scientific American many years ago, I encountered a peculiar binary sequence I describe below. Unfortunately I can't find a reference on this, so I would be grateful for any pointers or references.

Let $$\mathbb{N}$$ be the set of positive integers and let $$T = \{2^n: n\in \mathbb{N}\cup \{0\}\}$$ denote the set of powers of $$2$$. Let $$\text{m}:\mathbb{N}\to T\cup\{0\}$$ be defined by $$n\mapsto \max\big(\{0\}\cup \{t\in T: t.

We define $$a:\mathbb{N}\to\{0,1\}$$ recursively by

• $$a(1) = 1$$, and
• $$a(n) = 1-a(n-\text{m}(n))$$ for $$n\geq 2$$.

This sequence starts by $$10010110\ldots$$ and I recall that it has some peculiar properties such as, no non-empty finite sub-sequence occurs $$3$$ times in a row.

Question. Is $$\sum_{n=1}^\infty 2^{-n}a(n)$$ transcendent?

Let $$\{t(i)\}_0^\infty$$ be the Thue-Morse sequence. (It starts $$0,1,1,0,1,0,0,1,\ldots$$.).
I claim that your sequence is described by $$a(n)=1-t(n-1)$$ where $$n$$ is a positive integer. (It is similar to sequence A010059 in the OEIS, but its index starts at $$1$$ instead of at $$0$$.)
For $$n=1$$, $$a(1)=1-t(1-1)=1$$. We now consider $$n\ge 2$$.
From this OEIS link, let $$A_k$$ denote the first $$2^k$$ terms of $$t$$; then $$A_0=0$$ and for $$k\ge 0$$, $$A_{k+1}=A_k,B_k$$, where $$B_k$$ is obtained from $$A_k$$ by interchanging $$0$$'s and $$1$$'s. That is, $$1-t(i)=t(i-2^k)$$ where $$2^k\le i\le 2^{k+1}-1$$. Since $$\mathrm{m}(i)$$ is the largest power of $$2$$ less than $$i$$, then $$\mathrm{m}(i+1)=2^k$$. Thus, $$1-t(i)=t(i-\mathrm{m}(i+1))$$. Letting $$i=n-1$$ yields $$1-t(n-1)=t(n-1-\mathrm{m}(n))$$. Letting $$a(n)=1-t(n-1)$$ yields $$a(n)=t(n-1-\mathrm{m}(n))=1-(1-t(n-\mathrm{m}(n)-1))=1-a(n-\mathrm{m}(n))$$. $$\blacksquare$$
The Prouhet-Thue-Morse constant $$0.01101001\ldots$$ (in binary) (which is based on the Thue-Morse sequence) was shown to be transcendental by Kurt Mahler in 1929. It follows that the constant formed from your sequence is also transcendental.