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 $n\mapsto \max\big(\{0\}\cup \{t\in T: t<n\}\big)$.

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?